extreme value frequency analysis by tl-moments and

i

EXTREME VALUE FREQUENCY ANALYSIS

BY TL-MOMENTS AND TRANSMUTED DISTRIBUTIONS

By

Mirza Naveed Shahzad

Department of Statistics Quaid-i-Azam University, Islamabad,

Pakistan 2016

ii

EXTREME VALUE FREQUENCY ANALYSIS

BY TL-MOMENTS AND TRANSMUTED DISTRIBUTIONS

By

Mirza Naveed Shahzad

Supervised by

Dr. Zahid Asghar

A Thesis

Submitted to the Department of Statistics,

Quaid-i-Azam University, Islamabad

in fulfillment of the requirements for the Degree of

DOCTOR OF PHILOSOPHY

IN

STATISTICS

Department of Statistics Quaid-i-Azam University, Islamabad, Pakistan

2016

iii

CERTIFICATE

EXTREME VALUE FREQUENCY ANALYSIS BY TL-MOMENTS AND TRANSMUTED DISTRIBUTIONS

By

Mirza Naveed Shahzad (Reg. No. 03221011001)

A Thesis Submitted in the Partial Fulfillment of the

Requirements for the Degree of Doctor of Philosophy

We accept this dissertation as conforming to the required

standard

1.____________________ 2.____________________

(External Examiner) (External Examiner)

3.____________________ 4.____________________

Dr. Zahid Asghar Prof. Dr. Javid Shabbir

(Supervisor) (Chairman)

Department of Statistics

Quaid-i-Azam University, Islamabad, Pakistan 2016

iv

This dissertation is lovingly dedicated to

My parents, and all my brothers and sisters

who have raised me to be the person I am today

My wife who supported me much during this degree

My beloved kids: Asmara, and Farqleet

v

DECLARATION

I, Mirza Naveed Shahzad S/O Sardar Khan, Registration No. 03221011001, a student

of Doctor of Philosophy at Quaid-i-Azam University, Islamabad, Pakistan, do hereby

solemnly declare that the thesis entitled “Extreme Value Frequency Analysis by TL-

moments and Transmuted Distributions” submitted by me in partial fulfillment of the

requirements for Ph.D. degree in Statistics, is my original work and has not been

submitted and shall not, in future, be submitted by me for obtaining any degree from

this or any other University or Institution. And that to the best of my knowledge and

belief, this thesis contains no material previously published except where due

reference is made in the text of the thesis.

Date: ____________ Signature: ________________

(Mirza Naveed Shahzad)

vi

ACKNOWLEDGEMENT

This dissertation owes a great deal to the years of research that has been done

since I came to QAU for PhD degree. I worked with a number of distinguished

professionals whose contribution helped my research immensely. I would like to

express my gratitude to them.

In the first place, I am thankful to Almighty ALLAH for blessing me in

numerous ways to complete this dissertation.

I would like to pay my profoundest gratitude to Dr. Zahid Asghar for his

supervision, advice, and guidance from the very early stage of this research. He

provided me unflinching encouragement and support in various ways during my

doctoral research endeavor. His unique way of advising instilled in me the much

needed interest and enthusiasm to complete this research.

I express my gratitude to Prof. Dr. Javed Shabbir, Chairman, Department of

Statistics, Quaid-i-Azam University, Islamabad, for his valuable guidance and all

my teachers at Quaid-i-Azam University, Islamabad, who enlightened me for

years. Thanks to the office staff at the Department of Statistics, Quaid-i-Azam

University, Islamabad, for assisting me to complete the required office work and

for maintaining a good working atmosphere.

Thanks to all my friends and colleagues at University of Gujrat, Gujrat, proved

to be my ardent supporters. While it is impossible to name all, my grateful

appreciations are especially for Dr. Zahoor Ahmad, Dr. Fayyaz Ahmad, Dr.

Muqqadas Javed, Mirza Rizwan Sajid and Ms. Ummara Shahid for their

consistent encouragements.

Many thanks are due to Designation Prof. Dr. Balakrishnan, Narayanaswamy

for his invitation, guidance and help in exploring new research dimensions, during

my six months stay at McMaster University, Hamilton, Canada, financed by

International Research Support Initiative Program (IRSIP) of Higher Education

Commission (HEC) of Pakistan.

I want to extend my grateful acknowledgement to international reviewers,

external evaluators and anonymous referees of various journals who gave their

precious comments to enhance the quality of this thesis. My special thanks are

also extended to Dr. Faton Merovci and Mr. Henry So for their guidance in

compiling the R codes.

I would like to express my sincere thanks to special individuals like Amena

Arooj, Nadia Khan, Amjad Khan, Bishnu Katuwal, Ehsan Ullah, Yasser Abbas,

Dr. Farrukh Shehzad and Dr. Muhammad Nawaz for encouragement and support

for this research.

It is a pleasure to express my gratitude wholeheartedly to Dr. Zamir Hussain

for his moral support, guidance, cooperation and valuable comments in my

doctoral research are immeasurable.

vii

I am also very grateful to my bother Dr. Mirza Ashfaq Ahmad for his

encouragement, guidance and personal attention which have provided good and

smooth basis from admission to the completion of my doctoral research.

Last, but not least, I would like to thank my family, my parents who raised me

with love and supported me in all my pursuits, my brothers and sisters,

particularly, my wife and my kids Asmara and Farqleet for their love, patience,

and understanding, they allowed me to spend most of the time on this thesis.

I extend my gracious thanks to all well-wishers and motivators.

viii

ABSTRACT

The purpose of extreme value frequency analysis is to analyze past records of

extremes to estimate future occurrence probabilities, nature, intensity and frequency.

It is only possible if most suitable probability distribution is employed with proper

estimation method. Many probability distributions and parameter estimation methods

have been proposed in last couple of decade, but the quest of best fit has always been

of concern. In the continuity of this dimension, the fundamental aim of this

dissertation is to model the extreme events by proper probability distributions using

the most suitable method of estimation. This objective is achieved by reviewing and

employing the concept of L- and TL-moments and quadratic rank transmutation map.

The L- and TL-moments of some specific distributions are derived, and parameter

estimation is approached through the method of L- and TL-moments. In this study

three transmuted and two double-bounded transmuted distributions are developed and

proposed with their properties and applications. Moreover, the generalized

relationships are also established to obtain the properties of the transmuted

distributions using their parent distribution.

In the first part of the dissertation, it is observed that the Singh Maddala, Dagum, and

generalized Power function distribution are suitable candidates for extreme value

frequency analysis, as these densities are heavy-tailed in their range. In literature, the

theory of L- and TL-moments is considered best and extensively used for such

analysis. Therefore, the L- and TL-moments are derived, and the parameters of these

densities are estimated by employing the method of L- and TL-moments. These

estimation methods are compared with the method of maximum likelihood estimation

and method of moments using some real extreme events data sets. Simulation studies

have also been carried out for the same purpose. In these studies, superiority of the

method of L- and TL-moments has been justified.

In the second part of the dissertation, three heavy-tailed, flexible and versatile

distributions are introduced using the quadratic rank transmutation map to model the

extreme value data. The proposed distributions are the transmuted Singh Maddala,

transmuted Dagum and transmuted New distribution. The mathematical properties

ix

and reliability behaviors are derived for each of the proposed transmuted distribution.

The densities of order statistics, generalized TL-moments, and its special cases are

also studied. Parameters are estimated using the method of maximum likelihood

estimation. The appropriateness of the transmuted distributions for modeling extreme

value data is illustrated using some real data sets. The empirical results indicated that

the proposed transmuted distributions perform better as compared to the parent

distributions.

In literature, continuous double-bounded data is fairly popular. However, it is quite

unrealistic to analyze such kind of data using normal theory models. This type of data

is also targeted, and two new double-bounded distributions have been introduced, in

the third part of the dissertation. These developed distributions termed as transmuted

Kumaraswamy and transmuted Power function distribution. The most common

mathematical properties are derived, and it has been observed that the hazard rate

function have either increasing or bathtub shaped for these distributions. The method

of maximum likelihood estimation is employed for the parameter estimation and the

construction of the confidence intervals. The application and potential of these

distributions are investigated using real data sets. Comparatively, proposed double

bounded transmuted distributions performed better than their parent distributions in

real applications.

Finally, it has already been proved that transmuted distributions are better than their

parent distributions. But directly dealing with the transmuted density is complicated

and exhaustive especially for order statistics analysis. To make it simple, the

relationships between transmuted and parent distributions are established for the

single and product moments of order statistics. In addition, the generalized TL-

moments of the transmuted distribution and its special cases are derived using single

moments of the parent distribution. The established relationships are used for

parameter estimation, and a simulation study is also carried out to investigate the

behavior of the estimators. Moreover, the transmuted and parent distributions

relationships are illustrated through two well-known distributions and two real data

sets. Furthermore, it can be claimed on the base of established results; now it is quite

convenient to find the moments of order statistics, parameter estimates and especially

generalized TL-moments for transmuted distributions.

x

TABLE OF CONTENTS

CHAPTER 1 Introduction 1

1.1 Rational of study 5

1.2 Objectives of the study 6

1.3 Outline of Dissertation 7

CHAPTER 2 Review of Literature and Methodology of Extreme Value

Frequency Analysis 10

2.1 Parameter estimation 12

2.2 L-moments 12

2.3 TL-moments 14

2.4 Quadratic rank transmutation map 18

CHAPTER 3 Parameter Estimation by method of L- and TL-moments

for Extreme Value Analysis 21

3.1 Introduction 21

3.2 Parameter estimation of Singh-Maddala distribution 23

3.2.1 L- and TL-moments for Singh-Maddala distribution 24

3.2.2 L- and TL-moments ratios 26

3.2.3 Simulation Study 27

3.2.4 Application 28

3.3 Parameter estimation of Dagum distribution 33

3.3.1 L- and TL-moments for Dagum distribution 34

3.3.2 L- and TL-moments ratios 36

3.3.3 Simulation Study 37


3.4 Parameter estimation of generalized Power Function… 41

3.4.1 Method of Moments and moments ratios of the GPF… 43

3.4.2 L-moments and L-moment ratios of the GPF … 44

3.4.3 TL-moments and TL-moment ratios of the GPF … 45

3.4.4 Parameter estimators of GPF Distribution 46

3.4.5 Comparison of L- and TL-Moments by Simulation … 47


xi

3.5 Conclusion 56

CHAPTER 4 Extreme Value Analysis by Transmuted Distributions 57

4.1 Introduction 57

4.2 Transmuted Singh-Maddala distribution 59

4.2.1 Basic Properties 60

4.2.2 Reliability analysis 63

4.2.3 Order statistics of the transmuted Singh-Maddala … 66

4.2.4 Generalized TL-moments 67

4.2.5 Special cases of generalized TL-moment 68

4.2.6 Parameter estimation 70


4.3 Transmuted Dagum distribution 74

4.3.1 Basic properties 76

4.3.2 Quantile function and random data generation 78

4.3.3 Properties of TD distribution in term of reliability… 78

4.3.4 Order statistics of the transmuted Dagum distribution 81

4.3.5 TL-moments 82

4.3.6 Parameter estimation 83


4.4 Transmuted New distribution 88

4.4.1 Reliability analysis of the transmuted New distribution 90

4.4.2 Moments 90

4.4.3 Order Statistics 91

4.4.4 Estimation 92


4.5 Conclusion 96

CHAPTER 5 Extreme Value Analysis by Double Bounded Transmuted

Distributions 97

5.1 Introduction 97

5.2 Double bounded transmuted Kumaraswamy distribution 99

5.2.1 Basic statistical properties 102

xii

5.2.2 Reliability and hazard rate function 104

5.2.3 Order statistics of transmuted Kumaraswamy … 104

5.2.4 Generalized TL-moment and its special cases 105

5.2.5 Estimation and Information Matrix 107

5.2.6 Empirical Study 108

5.3 Double bounded transmuted Power function distribution 113

5.3.1 Mathematical properties 116

5.3.2 Quantile function and random number generation 118

5.3.3 Reliability analysis 118

5.3.4 Order statistics 120

5.3.5 Generalized TL-moment 120

5.3.6 Parameter Estimation 122

5.3.7 Monte Carlo Simulation study 123

5.3.8 Application of transmuted Power function distribution 124

5.4 Conclusion 128

CHAPTER 6 Relations between Transmuted and Parent distribution’s

Moments of Order Statistics 130

6.1 Introduction 130

6.2 Relation for single moments 132

6.3 Relation for product moments 134

6.4 Generalized TL-moment of the transmuted distribution 136

6.5 Relations for the single and product moments of transmuted … 138

6.5.1 Moments relations of Power and transmuted Power … 139

6.5.2 Moments relations of exponential and transmuted … 144

6.3 Parameter estimation 148

6.4 Application 150

6.4.1 Application of the transmuted Power function … 150

6.4.2 Application of the transmuted exponential distribution 151

6.5 Concluding remarks 153

xiii

CHAPTER 7 Summary, Conclusions and Recommendations 154

7.1 Parameter estimation by method of L- and TL-moments 154

7.2 New proposed Transmuted Distributions for Extreme Value … 155

7.3 Double Bounded Transmuted Distributions for Extreme … 158

7.4 Relationships between Transmuted and Parent distributions 159

7.5 Recommendations 159

REFERENCES 161

xiv

LIST OF FIGURES

Figure 3.1 Empirical and fitted cdf of Singh-Maddala distribution using

TL-moments estimates

30

Figure 3.2 Empirical and fitted cdf of Singh-Maddala distribution using L-

moments estimates

30


method of moments estimates

31


MLE estimates

31

Figure 3.5 PP-plots for all considered estimation methods for Singh-

Maddala distribution

32

Figure 3.6 Empirical and fitted cdf of Dagum distribution using TL-

moments estimates

39

Figure 3.7 Empirical and fitted cdf of Dagum distribution using L-moments

estimates

39

Figure 3.8 Empirical and fitted cdf of Dagum distribution using MoM-

moments estimates

40

Figure 3.9 Empirical and fitted cdf of Dagum distribution using MLE

estimates

40

Figure 3.10 PP-plots for all considered estimation methods for Dagum

distribution

41

Figure 3.11 The pdf’s of GPF distribution for various choice of parameters:

and ; , and

; , and ; 0,

10 and 2.0[2.0]10 with solid, dashed, dotted, dotdash and

longdash lines respectively

43

Figure 3.12 Bias of L- and TL-estimators by fitting the GPF distribution for

simulated data

48

Figure 3.13 RMSE of L- and TL-estimators by fitting the GPF distribution

for simulated data

49

Figure 3.14 Bias and RMSE of L- and TL-estimator β for GPF distribution

for simulated data

50

Figure 3.15 L-moment ratio diagram for and

curve 51

Figure 3.16 TL-moment ratio diagram for and

curve 51

Figure 3.17 Empirical, fitted cdf of Power Function distribution using TL-

moments estimates

54

Figure 3.18 Empirical, fitted cdf of Power Function distribution using L-

moments estimates

55

Figure 3.17 Empirical, fitted cdf of Power Function distribution using MoM

estimates

55

xv

Figure 4.1 The pdf’s of TSM distribution for various choice of parameters:

0.8[0.2]0.6; 2[1]7; 1.5[0.5]3.5;

3.0[0.5]5.0 and 1.5[0.5]3.5 with solid, dashed, dotted,

dotdash and longdash lines respectively

60

Figure 4.2 The cdf’s of TSM distribution for various choice of parameters:

0.8[0.2]0.6; 2[1]7; 1.5[0.5]3.5;

3.0[0.5]5.0 and 1.5[0.5]3.5 with solid, dashed, dotted,

dotdash and longdash lines respectively

61

Figure 4.3 The hazard functions of TSM distribution when 1 65

Figure 4.4 The hazard functions of TSM distribution when 2 65

Figure 4.5 Estimated densities and empirical histogram for the data set of

household expenditures

73

Figure 4.6 Empirical, fitted TSM and SM cdf for the data set of household

expenditures

73

Figure 4.7 The pdf’s of TD distribution for various choice of parameters:

3.0[1.0]7.0, 2.0[0.5]4.0, 0.5[1.0]4.5 and

1.0[0.5]1.0 with solid, dashed, dotted, dotdash and


75

Figure 4.8 The cdf’s of TD distribution for various choice of parameters:

3.0[1.0]7.0, 2.0[0.5]4.0, 0.5[1.0]4.5 and

1.0[0.5]1.0 with solid, dashed, dotted, dotdash and


76

Figure 4.9 The various shapes of reliability function for TD distribution 79

Figure 4.10 The behaviour of the hazard rate of TD distribution for various

parameters values such as: 1.0[0.5]1.0, 1.0[0.5]1.0, 0.75[0.25]1.75 and 0.8[0.1]-0.4,

0.75[0.25]1.75 and 0.2[0.2]0.8 with solid, dashed,

dotted, dotdash and longdash lines respectively

80

Figure 4.11 Empirical, fitted TD and Dagum cdf of the precipitation data set

and maximum distance highlight

86

Figure 4.12 PP-plots for fitted TD and Dagum distribution 87

Figure 4.13 The pdf’s of for various values of parameters:

1.0[0.5]3.0, 0.5; 2, 1.0[0.5]1.0 with solid,

dashed, dotted, dotdash and longdash lines respectively

89

Figure 4.14 The cdf’s of for various values of parameters:

1.0[0.5]3.0, 0.5; 2, 1.0[0.5]1.0 with

solid, dashed, dotted, dotdash and longdash lines respectively

89

Figure 4.15 Shapes of Reliability function of with various choices

of parametric values

90

xvi

Figure 4.16 The empirical and fitted cdfs of the life of fatigue fracture data 94

Figure 4.17 PP-plots of the fitted distribution of the life of fatigue fracture

data

95

Figure 4.18 The empirical and fitted cdfs of the life of maintenance data 95

Figure 4.19 PP-plots of the fitted distribution of the maintenance data 96

Figure 5.1 The pdf’s of TKw distribution for various choice of parameters:

0.5, 0.5,1[1]4.0, 0.1; : 2.5[1]6.5, 2.0[0.25]3.0, 0.8; : 2.0, 2.0[1]7.0, 0.5; :

3.0, 4.0, 1.0[0.5]1.0; with solid, dashed,

dotted, dotdash and longdash lines respectively

100

Figure 5.2 Figure 5.2: The pdf’s of TKw distribution for various choice of

parameters: 0.5, 0.5,1[1]4.0, 0.1; : 2.5[1]6.5, 2.0[0.25]3.0, 0.8; : 2.0, 2.0[1]7.0, 0.5; : 3.0, 4.0, 1.0[0.5]1.0;

with solid, dashed, dotted, dotdash and longdash lines

respectively

101

Figure 5.3 The mean plot of the TKw distributions with respect to the

parameters

102

Figure 5.4 Time series plot of Annual maximum peak flow at Kalabagh site 109

Figure 5.5 PP-plot, empirical, fitted TKw and Kw cdf for data set 1 111



Figure 5.8 The pdf of the TPF distribution for various values of the

parameters: 0.3, 0.5, 3.0, 5.0 and 0.0; 0.5 and

0.1[0.2]0.7; 3.0, 0.3[0.2]0.9; 0.5, 3.0,

0.5, 3.0 and 0.7[0.4]0.7 with solid, dashes, dotted and


115

Figure 5.9 The cdf of the TPF distribution for various values of the

parameters: 0.3, 0.5, 3.0, 5.0 and 0.0; 0.5 and

0.1[0.2]0.7; 3.0, 0.3[0.2]0.9; 0.5, 3.0,

0.5, 3.0 and 0.7[0.4]0.7 with solid, dashes, dotted and

longdash lines respectively.

116

Figure 5.10 The reliability functions of various values choices of parameters:

0.3, 0.7; 0.3, 0.7; 5.0, 0.7 and

5.0, 0.7 with solid, dashes, dotted and longdash lines

respectively

119

Figure 5.11 The hazard functions of various values choices of parameters 119

Figure 5.12 PP-plot of TPF distribution and PF distribution for

Communication transmitter failure data

126

Figure 5.13 Time series plot of annual maximum precipitation data of

Karachi, Pakistan

127

xvii

Figure 5.14 PP-plot of TPF distribution and PF distribution for precipitation

data

127

Figure 6.1 The pdfs of TPF distribution for various choices of the

parameters

140

Figure 6.2 The pdfs of transmuted exponential distribution for various

choices of the parameter

146

Figure 6.3 Empirical and PP-plot for fitting the TPF distribution 151

Figure 6.4 Empirical and PP-plot for fitting the transmuted exponential

distribution

152

xviii

LIST OF TABLES

Table 3.1 Summary of the average bias and MSEs of all the estimators ( ) of

the Singh-Maddala distribution for different sample size

28

Table 3.2 Parameter estimates of the SM-distribution using different

parameter estimation methods and the result of the goodness of fit

test

29

Table 3.3 C-, L- and TL-moments and moments ratios for monthly maximum

temperature of Jacobabad

32

Table 3.4 Summary of the average bias and MSEs of all the estimators ( ) of

the Dagum distribution for different sample size

37

Table 3.5 C-, L- and TL-moments and moments ratios for annual maximum

wind speed at Vancouver

38

Table 3.6 Parameter estimates of the Dagum distribution using different

parameter estimation methods and the result of the goodness of fit

test

38

Table 3.7 , and for three type of moments assuming varying

parametric values and for

52

Table 3.8 , and for type of three moments assuming varying

parametric values and for

53

Table 3.9 Parameter estimation using annual maximum precipitation data of

Karachi, Pakistan

54

Table 4.1 Summary Statistics of expenditure data, HIES, Pakistan 71

Table 4.2 Estimated parameters of TSM and SM distribution by MLE 72

Table 4.3 First four L-, TL-, LL- and LH-moments and moments ratios for

the household expenditure data

74

Table 4.4 Summary Statistics for monthly maximum precipitation data of the

Islamabad, Pakistan

85

Table 4.5 Estimated parameters of the TD and Dagum distribution for

precipitation data set

85

Table 4.6 First four sample moments, , and of C-moments, L- and

LT-moments for precipitation data set

87

Table 4.7 Parameter estimates, SE and CI of two data sets 93

Table 4.8 Goodness of fit measure for transmuted and parent distributions 94

Table 5.1 The estimates, standard error of estimates, confidence interval and

goodness fit criteria for three real data sets

110

Table 5.2 First four moments, , and of C-moments, L-, TL-, LL-

and LH-moments by three data sets

113

xix

Table 5.3 Estimates and MAE for different choices of parameter for TPF

distribution

123

Table 5.4 Estimates and MAE for different choices of parameter for TPF

distribution

124

Table 5.5 Estimated parameters with goodness of fit criterion of the TPF

distribution and PF distribution

125

Table 5.6 First four C-moments and TL-moments of Communication

transmitter failure data

126

Table 5.7 Annual maximum precipitation data of Karachi, Pakistan 127

Table 5.8 First four C-moments and TL-moments of annual maximum

precipitation data

128

Table 6.1 Means of PF distribution of order statistics for n = 1[1]10 143

Table 6.2 Means of TPF distribution of order statistics for n = 1[1]5 144

Table 6.3 Means of exponential distribution of order statistics for n = 1[1]12 147

Table 6.4 Means of transmuted exponential distribution of order statistics for

n = 1[1]6

148

Table 6.5 The average estimates with their corresponding MSEs with varying

sample size and parameters of the transmuted exponential

distribution

150

Table 6.6 First four L-, TL-, LL- and LH-moments for technical efficiency

scores data

151

Table 6.7 First four L-, TL-, LL- and LH-moments for military vehicles

failure data

152

1

CHAPTER 1

Introduction

The existence of extreme values in many areas has been witnessed particularly in

meteorology (extreme winds, heavy rainfall, heat waves, hurricanes, and droughts),

hydrology, finance (distribution of income, value-at-risk: maximal daily loss and re-

assurance), insurance, engineering, computer science, agriculture and medical

sciences. Therefore, accurate and reliable estimation of extreme events always

regarded as an area of high interest. As the advance knowledge of the strength and

magnitude of an extreme event, with a certain amount of accuracy can help in

minimizing the damages related to it. It is only possible when the forthcoming

extreme event is correctly quantified and properly estimated to take the precautionary

measures. Therefore, the estimation of extreme events has received much attention

in recent years and become a serious challenge for the researchers, scientists, and

analysts.

In our daily life, the usual interest is in the average characterization. Nevertheless,

rare/extreme events are far more crucial than the average one. Their occurrence is

responsible for massive destruction and loss of human lives. Consequently,

researchers, policy makers, medical and engineering professionals, etc. are always in

need to devise common strategies to provide maximum security from the disasters due

to extreme events. In recent years, the number of extreme events has been increased in

many parts of the world due to the exceptional heat waves, droughts, record-breaking,

floods, freezes and wind storms. The severity of extreme events is illustrated through

following few international and national examples.

A magnitude of 9.0 earthquakes became the causes of the tsunami with 33 foot

high waves that damaged the Tokyo, Japan on March 11, 2011. It caused extensive

devastation with officially confirmed 18,465 people killed/missed as reported by

2

National Police Agency of Japan. That tsunami had also affected nuclear meltdown

and hydrogen explosions Fukushima-I.

The wildfires in Russia, 2010 were the worst wildfires and its reason was associated

with high temperature. As in summer 2010, the highest temperature had been

recorded in Russian history. These wildfires created the smog, pollution, and heat

wave. As a result more than 56,000 people killed and extensively public health was

affected according to the Munich Reinsurance Company, German.

A catastrophic earthquake of a magnitude of 7.0 struck Haiti on January 12,

2010. Three million people were affected and from 100,000 to 160,000 were killed by

the quake. According to the estimation of government of Haiti, 30,000 commercial

and 250,000 residential building were severely damaged or collapsed (source, Cavallo

et al., 2010 and Wikipedia). And massive snowfall in December 2007 covered Central

and Eastern Canada, which shut down many cities for several days.

On October 8, 2005, a severe earthquake struck in Pakistan in which estimated

86,000–87,351 people dead, 69,000–75,266 injured and 2.8 million displaced. This

earthquake affected more than 500,000 families, and more than 780,000 buildings

were either destroyed or damaged. It also affected some adjacent parts of Afghanistan,

India and China. The fatalities and destruction resulted from this earthquake had made

it one of the most destructive earthquakes (for detail see, USGS, 2009 and Hussain et

al., 2006) in recent memories of the region.

In 2010, Pakistan had faced the most disastrous flood in the history of the country.

This flood destroyed approximately 1.6 million homes and affected 20 million people.

It also left 14 million people without homes, infrastructure, crops and millions

vulnerable to malnutrition and waterborne disease. The total economic impact of this

flood was estimated U.S. $43 billion, approximately, (Hicks and Burton, 2010 and

Singapore Red Cross, 2010).

In Mohenjo-Daro, Sindh, Pakistan, the highest and hottest temperature ever recorded

was 53.5 °C (128.3 °F) on 26 May 2010 (Ali, 2013). It was the hottest temperature

consistently measured on the continent of Asia and the fourth highest temperature

ever recorded on earth (Pakmet, 2010). Currently in June 2015, due to an extremely

high temperature in Karachi, Pakistan more than 2000 people died due to the

dehydration and heat stroke (Imtiaz, 2015).

3

According to the WHO report, the wave of dengue fever in 2011 struck the eastern

province of Punjab, Pakistan. Mortality from Dengue fever was at least 365 people.

There were 21,597 laboratories confirmed and 2,52,935 suspected cases had been

reported which made it the world’s biggest outbreak of dengue fever ever (Shakoor,

2012).

Another example of the extreme event from Pakistan is the extraordinary drought of

1998–2002, which is considered the worst in 50 years. Western and central parts of

Balochistan Province remained in the grip of this drought for about five years.

Finally, we refer the United Nations University for Environment and Human Security

(UNU-EHS) report. The UNU-EHS has measured the natural disaster risk in 2015 and

ranked 173 countries based on their disaster risk. According to this report, Pakistan

has a 7.21% risk with rank 101 and Pakistan fall in the fourth risky category out of

five.

The above-listed examples are only a few, and these are related to the extreme in

nature. There are many other applied or practical situations that are mainly concerned

with extremes, including minimum component breaking stress or minimum strength

of materials, maximum wind velocity, maximum traffic at a peak hour, maximum

daily loss might occur in the stock market, maximum claim size of an insurance

company and so on. Consequently, it is revealed that an accurate and reliable

estimation of the extreme events is deemed essential to get most suitable and reliable

statistical analysis to deal with such extreme conditions.

There are four main approaches to analyzing the extreme values, such as deterministic,

parametric, stochastic and probabilistic approach. The analysis by deterministic

approach uses the defined chemical and physical rules. The recorded data at different

locations and times are often analyzed by the parametric approach. Stochastic

approach analyzes the extreme values using time-series methods by considering their

frequency and time of occurrence. Probabilistic approach measures the pattern of the

data and uncertainty by using probability distributions. Therefore, it is the only

reliable source required to incorporate the effects of such phenomena into decisions

(WMO, 2009). The methodology of this approach takes into account the full range of

the observations with their magnitude. Therefore, it provides a more realistic basis for

quantifying and taking decisions about uncertainty. Tung and Yen (2005) also

4

indicated that the uncertainty is very important for indicating the reliability of an

estimate in extreme value problems. The degree of uncertainty depends on return

period, sample size and the underlying probability distributions.

A wide variety of probability distributions were compared in various applications in

extreme value analysis, for example, Hosking and Wallis (1997) analyzed the flood

data by uniform, exponential, Gumble, Normal, generalized Extreme-Value,

generalized Pareto, generalized Logistic, Pearson type III, Lognormal, Kappa, and

Wakeby distribution. Zalina et al. (2002) considered eight candidate distributions,

including Gamma, generalized Pareto, generalized Normal, generalized Extreme-

Value, Log Pearson Type III, Pearson Type III, Gumbel, and Wakeby distribution to

model the annual maximum rainfall estimates for Malaysia. Husak et al. (2007) used

Gamma distribution to forecast the monthly maximum rainfall in Africa. Morgan et

al. (2011) modeled the offshore wind speeds by Weibull, bimodal Weibull, Kappa,

Wakeby and Lognormal distributions. Buchholz (2013) found that the Weibull and

Gumbel distribution were the best for modeling the traffic speeding data. There are

various studies those were investigated the extreme events using more or less similar

number of distributions to observe their intensity and probability of occurrence.

Besides these popular models, many other distributions are also studied for extreme

events because the selection of distribution depends on the nature of available data

and the problem at hand. As some extreme events data sets tend to be skewed or

heavily skewed. Therefore, in every analysis, some suitable candidate distributions

are compared and reviewed to take the decision regarding truly fitted distribution.

Probability distributions are often used to describe the data, as probability distribution

presents a smooth and consistent interpretation of the data, provide more accurate

statistical information, including, quantiles and provide realistic range of the random

variable that it may assume.

According to Hosking (1992), L-moments characterize the distributional shape,

preferably better than the conventional moments. Asquith (2007) narrated that the L-

moments are better for heavy-tailed distributions than conventional moments. These

L-moments methodology has extensively been used by hydrologists (Chen et al.,

2006). In 2003, Elamir and Seheult introduced Trimmed Linear (TL) moments as a

modified form of L-moments. Both L- and TL-moments are more robust towards the

outliers. Therefore, L- and TL-moments based extreme value frequency analysis is

5

practiced in various studies across the world. These moments are also used for the

estimation of parameter, and most of the extreme events analysts used the method of

L- and TL-moments and developed these estimation procedures for some densities. In

this dissertation, we have derived the L- and TL-moments and estimation procedure

for three parameters Singh-Maddala, Dagum, generalized Power function distribution

for the extreme value frequency analysis.

Hosking and Wallis (1997) also observed that the distributions having two parameters

produced biased results in comparison to the distribution having three to five

parameters especially in estimating extreme upper tail of the distribution.

Additionally, no particular distribution or class of distributions is appropriate for all

extreme event studies. The analysts and researchers are still in the quest to find out the

best models for such analysis. The current study introduced and proposed five new

candidate distributions with three and four parameters for the estimation of extreme

events assuming the quadratic rank transmutation map (QRTM). Shaw and Buckley

(2009) introduced this map, and it embeds an additional parameter in parent density to

generate more flexible and versatile density and distribution function.

1.1 Rational of study

An extreme event, like floods, rainfall, drought, tsunami, very high rise or fall in the

stock market, etc., is an inevitable phenomenon that occurs from time to time in

various parts of the world. Their occurrence results in the loss of infrastructure,

money and deaths of living beings. Therefore, the public often demands protection or

timely information to make necessary arrangements. However, protective measures

require accurate and reliable estimation of frequency and magnitude of such events.

So interest is in the modeling of extreme behavior of natural phenomena rather than

its average behavior. Almost all engineering designs heavily depend on extremes

because largest and smallest values are the main parameters that lead to failure of

engineering work. So the knowledge of probability distributions for the maxima and

minima are important to observe the pattern and prediction of the relevant

phenomena. This estimation and prediction are prudent to quantify and design the

engineering structures for achieving the balance between cost and safety goals.

Therefore, it is useful to develop the models those can cater such accurate estimation.

So specifically, our study is designed to find the best estimation method to estimate

6

the parameters for some models and to introduce five new generalized probability

distributions for extreme value analysis. The generalization of the distributions is

approached by the QRTM and termed as transmuted distributions. Transmuted

distributions provide a better fitting for extremes, but to find the properties of the

transmuted distributions comparatively laborious or difficult. Therefore, relationships

are developed between transmuted and parent distribution for single order and product

order moments. The relationships are also established to find the L- and TL-moments

of the transmuted distribution using the single order moments of the parent

distribution.

1.2 Objectives of the study

The objective of this research is three-fold with a special focus on the extreme value

frequency analysis. First is to introduce and compare the parameter estimation

methods for some distributions. Second is to develop some new transmuted

distributions with their properties, parameter estimation and applications and third is

to establish the relationships between transmuted and parent distribution. The specific

objectives of the study are stated as follows

Derivation of L-moments, TL-moments and parameter estimation for the

Singh-Maddala, Dagum, generalized Power function distribution through the

method of L-moments, method of TL-moments and method of moments.

Comparison of the moment’s estimation methods with maximum likelihood

estimation (MLE) to obtain the precise and accurate estimation method for the

analysis of the extreme events.

Development of some new distributions for extreme value analysis, such as

transmuted Singh-Maddala, transmuted Dagum and transmuted New

distribution with their properties and applications.

Development of the bounded distribution to model the extreme value data, like

transmuted Power function and transmuted Kumaraswamy distribution with

their properties and applications.

Establishment of the relationships between transmuted and parent

distributions. To avoid the complex mathematical derivation and to obtain the

properties of the transmuted distribution conveniently using the properties of

the parent distribution.

7

This study widened the application of the Singh-Maddala, Dugam and generalized

Power function distribution for the extreme value frequency analysis. Along with this

parameters estimate of these distributions are obtained by the method of moments,

method of L-moments and method of TL-moment. In this way, it is very useful for

statisticians and extreme events analysts to apply the most suitable method of

estimation to get true trend and prediction of the extreme events. Moreover, we have

introduced the five new flexible distributions, which can model the extreme events

more accurately. It will be a worthwhile contribution in the literature and these

distributions will provide better models to study extreme events than their parent

distributions. Furthermore, this study has also addressed the difficulties to derive the

order statistics, L-, TL-, LL- and LH-moments of the transmuted distribution.

Therefore, to deal with single moments, product moments and generalized TL-

moments the relationships are introduced. Now it has become easy to get order

statistics, L-, TL-, LL- and LH-moments of the transmuted distribution using the

established relationships.

This study will serve a useful purpose to statisticians, applied mathematicians,

engineers, meteorologist, medical practitioners and hydrologists. Especially, those are

engaged in the research work of the extremes and want to know their specified

probability distribution and accurate parameter estimates to measure the amount of

uncertainty and risk of next extreme event to inform management for their planning.

1.3 Outline of Dissertation

This dissertation consists of seven chapters and is structured as follows.

Chapter 1 presents the background, motivation, highlights the scope and objectives of

the study.

Chapter 2 describes few recent applications of extreme value frequency analysis. We

comprehensively review the L- and TL-moment with their method of parameter

estimation which shows the importance and applicability of the procedure. It also

introduces the QRTM to develop the flexible and versatile transmuted distributions.

Chapter 3 presents the derivation of the conventional, L- and TL-moments and

moment ratios for the Singh-Maddala, Dagum and generalized Power function

distributions. The parameters of these distributions are estimated using the method of

8

moments, method of L- and TL-moments and compared with the method of MLE

assuming a comprehensive simulation and real data studies. The work presented in

this chapter partially appeared in Shahzad1 and Asghar

2 (2013a) and Shahzad

1 and

Asghar2 (2013b).

Chapter 4 generalizes parent distributions by the QRTM and proposes the transmuted

Singh-Maddala, transmuted Dagum and transmuted New distribution. The graphical

presentations justify the flexibility and versatility of the transmuted distributions.

Basic mathematical properties, order statistics and L-, TL-, LL- and LH-moments are

derived for these distributions. Parameter estimation approaches through maximum

likelihood, and Newton-Raphson estimation techniques and proposed distributions are

compared with the parent distributions. The utility and potentiality of the proposed

models are illustrated using real data sets of extremes. Some contents of this Chapter

are published in Shahzad1 and Asghar

2 (2015a).

Chapter 5 deals with the two-sided bounded distributions as it is proved that bounded

distributions model the hydrology, meteorology, social and behavioral sciences data

more accurately than unbounded distributions. To enhance the flexibility and

versatility of the bounded distribution, QRTM has been used and proposed the

transmuted distributions like transmuted Power function and transmuted

Kumaraswamy distribution. The properties and estimation of parameters have been

studied for these distributions as in Chapter 4. Some interesting features of transmuted

Power function distribution are accepted in Shahzad1 and Asghar

2 (2015b).

Chapter 6 establishes the relationship between the transmuted and parent

distributions. To deal directly with the transmuted density is complicated and

exhaustive. Therefore, for connivance, the relationships are established for the single

and product moments of order statistics. In addition, the generalized TL-moments of

the transmuted distribution and its special cases are derived from single moments of

the parent distribution. The established relations are also used for parameter

estimation, and a simulation study is also carried out to investigate the behavior of the

estimators. Finally, the relationships between transmuted and parent distributions are

illustrated through two well-known distributions and two real data sets.

1 Mirza Naveed Shahzad, author of this dissertation

2 Zahid Asghar, Supervised this dissertation

9

Chapter 7 summarizes the major results and conclusions of this study. Some

recommendations for future work are also included at the end of this chapter.

10

2. CHAPTER 2

Literature and Methodology related to Extreme Value

Frequency Analysis

The purpose of analyzing historical extreme events is to predict their nature, intensity

and frequency. Therefore, the analysis of extreme events has been a dimension of

interest since a long time ago. Initial contributions in this analysis include the work of

Horton (1913), Fuller (1914), Hazen (1921), Foster (1924) and Fisher and Tippett

(1928). Later on, the contribution of the Gumbel (1941, 1954) and Jenkinson (1955) is

also considered as the foundation for extreme value frequency analysis. Vrijling and

Van-Gelder (2005) added that many studies have been carried out since 1920 on safety

and risk analysis, especially in the area of extreme value statistics. Gumbel (1958)

was the first who developed a set of new limiting distributions for extreme value

frequency analysis. Originally frequency analysis was approached by three distinct

probability distributions (log-normal, Gumbel and Pearson Type-III) and the

estimation of parameters was approached either by the plotting positions or method of

moments. Occasionally method of Maximum Likelihood Estimation (MLE) was also

employed. Currently, L- and TL-moments are more popular for extreme value

frequency analysis and method of L- and TL-moments for estimation of the

parameters of the probability distributions.

Extreme value analysis or extreme value theory is the branch of statistics that used to

find the properties of extreme events. It helps to fit theoretical distribution(s) on the

data, to describe the behavior of extreme values through summarization and

estimation of the parameters. Mathematically the extreme value theory is the study of

statistical formation and behavior of the data series, like

{ }

or

{ }

11

where { } is a sequence of independent random variables those follows

common probability distribution function. In real time applications, variables

represent values measured on a fixed regular time interval, for example annually,

quarterly, monthly, weekly, hourly or any other specified time period. It can easily be

understood through the following example. Record the daily temperature of a site for

years, then collect the maximum value from each year, in this way we have data

points. Now asymptotically consider , then there is a family of distributions

that is applicable for modeling the extreme in natural phenomena.

Initial applications of the extreme value frequency analysis were largely found in the

area of hydrology. Currently, almost every phase of meteorology, hydrology,

environmental and natural sciences is subjected to frequency analysis. There is a vast

amount of literature available on extreme value frequency analysis with a wide range

of applications in different fields. For instance, in insurance and finance (Gilli, 2006;

Paul et al., 1997), risk management (Marimoutou et al., 2009; Liu, 2013; Ayyub,

2014), ocean wave modeling (Stansell, 2005; Moeini et al., 2010; Hosking, 2012),

alloy strength prediction (Tryon & Cruse, 2000), memory cell failure (McNulty et al.,

2000), management strategy (Dahan & Mendelson, 2001; Wang et al., 2008),

hydraulics engineering (Morrison and Smith, 2001; Katz et al., 2002; Klein et al.,

2009; Horritt et al., 2010; Wehmeyer et al., 2012), wind engineering (Harris, 2001;

Walshaw, 2000; Lombardo et al., 2009; Hundecha et al., 2008; Anastasiades and

McSharry 2014), precipitation modeling (Feng et al., 2007; Maraun et al., 2009;

Westra et al., 2013; Du et al., 2014; Junqueira et al., 2015), structural engineering

(Zidek et al., 1979; Grigoriu 1984; Crespo-Minguillón and Casas, 1997), biomedical

data processing (Roberts, 2000; Kalbfleisch and Prentice, 2011), electrical related

matter (Nelson, 2004; Byström 2005; Klüppelberg et al., 2010; Lawless, 2011; Bunn

et al., 2013), assessment of meteorological change (Thompson et al., 2001; Ferro and

Segers, 2003; Cooley, 2009; Coumou and Rahmstorf, 2012; Kharin et al., 2013;

Keellings and Waylen, 2014), modelling volcanic magnitudes (Coles and Sparks,

2006); analysis of advanced age mortality (Watts et al., 2006), thermodynamics of

earthquakes (Lavenda and Cipollone, 2000; Pisarenko and Sornette, 2003; Pisarenko

et al., 2014; Shin, 2015), pollution studies (Smith, 1989; Ercelebi and Toros, 2009;

Reich; 2013), food science (Kawas and Moreira, 2001; Kumar and Chatterjee, 2005;

Hamed and Rao, 2010; Hussain, 2011; Smithers et al., 2015) and world records in

12

sports (Einmahl and Magnus 2008; Einmahl and Smeets, 2011; Henriques-Rodrigues,

2011). These are few application and studies those analyzed extreme value data by

statistical approach especially using probabilistic models. As a conclusion, the

application of extreme value analysis is observed in all areas of human life.

2.1 Parameter estimation

The main purpose of extreme value frequency analysis is to measure the intensity of

extreme events related to their frequency of occurrence through the use of probability

distributions. It is one of the oldest and most frequent uses of probability theory in

every field where extreme events observed. The next obvious step is to estimate the

parameters of the considered probability distribution(s). In extreme value analysis,

various parameter estimation methods are in practice while fitting probability

distributions. The well-known methods are method of sextiles (Jenkinson, 1969),

method of MLE (Jenkinson, 1969, Prescott and Walden 1980, 1983), method of

Probability Weighted Moments (PWM) (Hosking et al., 1985), method of L-moments

(Hosking, 1990), method of LH-moments (Wang, 1997), method of LL-moments

(Bayazit and Onoz, 2002) and method of TL-moments (Elamir and Seheult, 2003). In

the last few years the L- and TL-moments and parameter estimation by these

moments gain fair popularity in the field of frequency analysis.

2.2 L-moments

L-moments and parameter estimation through the method of L-moments were

introduced by Hosking (1990). L-moments are an attractive alternative than the

conventional moments (C-moments) as these are the expectations of linear

combinations of order statistics. These moments exist for all random variables whose

mean can be defined in close form. L-moments describe the geometry of the

distributions like other statistical moments with a similar interpretation. According to

Hosking (1992), L-moments characterize the distributional shape preferably better

than the C-moments. The parameter estimation of heavy-tailed distributions through

the method of L-moments is much better than the method of moments (Asquith,

2007). In addition, the properties of L-moments hold in a wide range of practical

situations. L-moments also give asymptotic approximations to sampling distributions

better than the C-moments and provide better identification of the parent distribution

13

that a particular data actually have (Ariff, 2009). Furthermore, L-moments are less

sensitive to outliers (Vogel and Fennessey, 1993).

Mathematically L-moments are defined as, let be a sample of size

with probability density function (pdf), cumulative density function (cdf) and

quantile function, , and , respectively. Then

denote the corresponding order statistics. Hosking (1990)

proposed the population L-moments as a linear combination of PWM and defined the

th L-moment as follows

∑ (

)

(2.1)

The expression of the expected value of the th order statistics of the random sample

of size is defined as

∫ [ ] [ ]

(2.2)

This expression can also be expressed using quartile function, such as

∫ [ ] [ ]

(2.3)

Substituting in expression (2.1) we obtained

∑ (

)

∫ [ ] [ ]

(2.4)

or

∑ (

)

∫ [ ] [ ]

(2.5)

It is easy to establish the expression for a particular order of L-moments using (2.4) or

(2.5).

It is proved that the first two L-moments represent the location and variability

characteristic of the data, respectively. On the basis of the first two L-moments, the

coefficient of variation of L-moments that is analogous to the classical coefficient of

variation is defined as

⁄ . (2.6)

The higher ordered L-moments ratios of a random variable are defined as

14

⁄ (2.7)

L-moments ratio (L-skewness) tells us about the asymmetry and (L-kurtosis)

indicates about the ratio of the peakedness of a probability distribution alike the

skewness and kurtosis from the C-moment ratios. The summarization of the

probability distribution through L-moments measures provides more accurate

information than the conventional measures.

Let be the ordered sample then the th sample L-moment

( ) defined by Hosking (1990) is defined as

( )

∑ ∑ ∑

∑ (

)

(2.8)

Thus, from (2.8) we can find the sample L-moments corresponding to the for

Hosking et al. (2005) showed another way to calculate the , that is as

follows

∑

(2.9)

Sample L-moments are used to summarize the basic properties of the sample

distribution in the same fashion as sample C-moments. L-moments are robust and less

sensitive than C-moments to sampling variability and to outline in the data (Hosking,

1990). Therefore, parameter estimation through L-moments provides more accurate

estimates (Bílková, 2014).

2.3 TL-moments

Elamir and Seheult (2003) introduced some robust modification in L-moments and

proposed TL-moments, which is a generalization of L-moments. TL-moments

overcome the problems of L-moments such as L-moments are sensitive towards the

lower parts, assign more weight to large sample values of the distribution and only

possible if the distribution has a finite mean. TL-moments are more robust towards

outliers than L-moments. TL-moments provide the opportunity to trim any possible

the number of observations from the data. However, herein we focus only on the

symmetric case.

Elamir and Seheult (2003) defined the th TL-moment of the random variable as

15

∑ (

)

(2.10)

Here, is assumed to trim only one extreme value from both sides, therefore,

TL-moments when , have the following form

∑ (

)

(2.11)

It is easy to obtain the particular TL-moments for specific value of . The first two

TL-moments define the location and variability of the probability distribution. TL-

moments ratios are the analogues of the C- and L-moments ratios. These moment

ratios, coefficient of variation , coefficient of skewness and coefficient of

kurtosis are given as

⁄ , (2.12)

⁄ (2.13)

and

⁄ (2.14)

respectively. Where TL-skewness and TL-kurtosis are denoted by

and

,

respectively.

Elamir and Seheult (2003) defined the sample TL-moments

, corresponding to the

population L-moments

as follows

∑ [

∑ (

) (

) (

)

(

)

]

(2.15)

Thus, from (2.15) the particular sample TL-moments corresponding to population TL-

moments can be obtained by substituting different values of . These sample

moments are used to obtain the sample TL-moments measures such as location,

variability, , and .

In several studies, estimation of parameters has been done using the method of L- and

TL-moments to determine the most suitable fitting of the probability distribution(s) on

the original data. Few studies related to parameter estimation by employing method of

L- and TL-moments have been completed in the last two decades. Some of them are

summarized as below.

16

The methodology of L-moments and method of L-moments for the parameter

estimation first time developed by Hosking (1990). In his study, he derived L-

moments and parameters estimators of the Uniform, Exponential, Gumble, Logistic,

Normal, generalized Extreme-Value, generalized Pareto, generalized Logistic,

Lognormal and Gamma distribution by the method of L-moments. Elamir and Seheult

(2003) introduced TL-moment and the parameters estimation of the Normal, Logistic,

Exponential and Cauchy distribution by the method of TL-moments.

Asquith (2007) derived the first five L- and TL-moments and estimated the

parameters of 4-parameter generalized Lambda distribution employing the method of

L- and TL-moments. This distribution is a flexible distribution and capable for heavy

tails data such as extreme events data. The simulation study showed that TL-

moments are more robust in the presence of high outliers.

The parameter estimation of the generalized Pareto distribution is approached through

the method of L- and TL-moments (Abdul-Moniem, 2009) and showed by simulation

study that the method of TL-moments provides the smaller mean square error than the

method of L-moments. In this study first four L- and TL-moments are derived to

obtain L-skewness, TL-skewness, L-kurtosis and TL-kurtosis for Generalized Pareto

distribution.

The generalized Logistic distribution has been implemented extensively in

hydrological risk analysis and in extreme events evaluation (for instance, Lim and

Lye, 2003 and Ashkar et al., 2006). Ahmad et al. (2011) introduced TL-moments of

the generalized Logistic distribution and compared the method of L- and TL-moments

with Method of Moments (MoM) for its parameter estimation. They concluded that

the performance of the method of TL-moments is better than the method of L-

moments and MoM through simulation study and stream flows data of Terengganu

Station, Malaysia.

Shahzad and Asghar (2013a, 2013b) estimated the parameters of the Singh-Maddala

and Dagum distributions, respectively, employing method of L- and TL-moments.

They derived first four L- and TL-moments including L- and TL-skewness and L- and

TL-kurtosis. The results of the parameters estimation revealed that TL-moment

estimators are less biased and have smaller root mean square errors than the method

17

of L-moments and MoM. These results hold for all possible parametric values and

sample sizes used in Monte Carlo simulation study.

Asquith (2014) studied the parameter estimation of the 4-parameter Asymmetric

Exponential Power distribution using the method of L-moments. Algorithms for

parameter estimation are suggested in the R-language environment for statistical

computing. Algorithms were written to calculate L-moments and to provide reliable

parameter estimation of this distribution. Finally, the application of L-moments

estimation provided using slight asymmetric and heavy-tailed datasets.

Erisoglu and Erisoglu (2014) proposed parameter estimation of mixture distributions

through the method of L-moments first time. They estimated five parameters of the

two component mixture of Weibull distributions. The proposed method showed more

flexibility and potentiality than the method of MLE with respect to bias, the mean

total error, the mean absolute error and completion time of the algorithm by

simulation and real data study.

Most recently, Shahzad et al. (2015) conducted a study for the parameter estimation

of Power Function distribution with L- and TL-moments. They derived L-moments,

TL-moments, LL-moments and LH-moments of Power function distribution. In

addition, the population L- and TL-skewness and L- and TL-kurtosis have also been

derived. Parameters of the density are estimated using the method of L- and TL-

moments and compared with MoM and method of MLE on the basis of bias, root

mean square error and coefficients through a simulation study. Finally, the conclusion

was in the favor of method of L-moments, and it was equally valid for different

parametric values and sample size.

Some other studies have also used the method of L- and TL-moments for parameter

estimation of the exponential (Abdul-Moniem, 2007), exponentiated Pareto (Ashour

et al., 2015), generalized Rayleigh (Kundu and Raqab, 2005), generalized Pareto

(Abdul-Moniem and Selim, 2009), generalized Lambda (Karvanen and Nuutinen,

2008), family of Dagum (Pant and Headrick, 2013), Burr Type VII distributions (Pant

and Headrick, 2014), etc.

18

2.4 Quadratic rank transmutation map

To model the extreme events, numerous probability distributions have been

introduced in the literature. Still no particular class of distributions or distribution is

considered superior for all extreme events studies. The analysts and researchers are

still in the quest to find out the good models. In this study, the Quadratic rank

transmutation map (QRTM) has been used to obtain the transmuted distribution, in

order to generate more flexible and versatile distribution function to model the

extreme value data. In this context, transmuted distributions are considered to be more

useful than their own parent distributions. Transmuted distribution embeds an

additional parameter in parent density to generate transmuted density using QRTM.

The QRTM is a special case of the general rank transmutation mapping, which is

defined by Shaw and Buckley (2009) without loss of generality. Suppose that there

are two probability distributions with a common sample space and having the

cumulative distribution functions and , then

{ }

{ },

where and are the quantile function (inverse cumulative distribution

functions).

The two functions and map the unit interval [ ] into itself, and

under suitable assumptions are mutual inverses. They satisfy and

This rank transmutation should be continuously differentiable otherwise

discontinuity may occur in the transmuted density. Now the quadratic rank

transmutation is defined for | | as,

which become the base of the cdf of the transmuted probability distribution, and this

cdf is defined as follows

[ ( )] (2.16)

which yields probability density function on differentiation,

[ ( )] (2.17)

Where and are the probability density and cumulative distribution

functions of the parent distribution, respectively.

19

The parameter lies between [ ], and the extreme values of the produce two

extreme cases. In particular, produces and

generates [ ]. It is also observed that when then the cdf

and pdf of the transmuted distribution subsidence to the parent distribution.

The procedure of QRTM is used to obtain the transmuted distribution. Transmuted

distributions are more flexible and versatile than the parent distribution and able to

model a variety of real data. Some of the recent work are summarized below.

Ashour et al. (2013) introduced the transmuted Lomax distribution and in the same

year Ashour et al. (2013) developed the transmuted Exponentiated Lomax

distribution. The basic statistical properties, reliability analysis and order statistics for

both the distributions were derived. The parameters of the densities were estimated by

the method of MLE and concluded that the transmuted densities are a better

alternative to the other models for modeling positive real data.

Aryal and Tsokos (2011) used the QRTM to develop the transmuted Weibull

distribution. Comprehensive descriptions of statistical properties along with reliability

behavior of the distribution were studied. The parameter estimation incorporated

through the method of MLE. The practicality of the transmuted Weibull distribution

was illustrated using breaking stress of carbon fibers and tensile fatigue characteristics

of a polyester data and proved the superiority of this new density. Elbatal and Aryal

(2013) extended this work and proposed transmuted additive Weibull distribution and

almost fourteen distributions became its special case. The shape of the hazard

function is a bathtub and due to this property, it became attractive for lifetime

applications.

Due to the popularity and applicability of the Weibull distribution, Elbatal (2013)

presented the transmuted modified inverse Weibull distribution, Ashour and Eltehiwy

(2013) proposed the transmuted Exponentiated modified Weibull distribution, Khan

and King (2012) introduced transmuted generalized Inverse Weibull distribution,

Khan and King (2013) generated transmuted modified Weibull distribution, Ahmed et

al. (2014) introduced transmuted complementary Weibull Geometric distribution,

Merovci and Elbatal, (2014) proposed transmuted Weibull Geometric distribution and

recently, Saboor et al. (2015) introduced transmuted Exponential Weibull

20

distribution. All these authors applied the analytical results of these transmuted

distributions to model real world data and obtained better fitting results.

Merovci (2013) introduced a new generalization of the Rayleigh distribution called

the transmuted Rayleigh distribution and this generalization was generated by using

QRTM taking the Rayleigh distribution as the parent distribution. Merovci (2013)

generated the transmuted generalized Rayleigh distribution and Ahmad et al. (2014)

proposed the transmuted Inverse Rayleigh distribution. In these studies

comprehensive description of statistical properties of the proposed distributions along

with their reliability behavior and order statistics were provided. The usefulness of

these transmuted distributions for modeling data is illustrated using real data.

Most recently, Shahzad and Asghar (2015) proposed new generalization of Dagum

distribution using QRTM and proposed transmuted Dagum distribution. In this study,

various popular properties of the proposed density have been studied. Additionally,

the reliability and hazard rate functions are derived and presented graphically. The th

order and joint order statistics of the proposed distribution obtained along with TL-

moments with its special cases. Shahzad and Asghar (2015) have introduced the

generalized TL-moments for transmuted distribution first time. The parameters

estimation of the transmuted Dagum distribution approached through method of

MLE. The real data of monthly maximum precipitation data of Islamabad, Pakistan is

used to check the fitting of the proposed model and they concluded that the

transmuted Dagum distribution is more flexible and more appropriate than that of

Dagum distribution.

21

3. CHAPTER 3

Parameter Estimation by method of L- and TL-moments for

Extreme Value Analysis

3.1 Introduction

Modeling, accurate inference, and prediction of extreme events are very important in

every field to minimize the damage due to extremes as much as possible. Probabilistic

models secure a useful purpose to model and predict such extreme events. And the

fitting of probabilistic models heavily depends on parameter estimation method, as

every method of estimation is not suitable for every probability distribution. The

method of MLE is commonly used for parameter estimation but the extreme value

data are frequently heavy-tailed data and in this situation method of L- and TL-

moments are considered alternatively. Method of L- and TL-moments usually

provides robust results in the field of extreme value analysis (e.g., see, Asquith, 2007,

2014 and Shahzad and Asghar 2013, 2015).

In this chapter, parameter estimation methods such as the method of L- and TL-

moments are developed for some specific heavy-tailed distributions. These estimation

methods have been compared with the MoM and MLE to find out the most reliable

and accurate estimation method for extreme value data analysis. L- and TL-moments

and their parameter estimation approach also provide the precise results with the good

summarization and description of the observed sample data. Therefore, we estimated

the parameters of the Singh-Maddala, Dagum and generalized Power function

distributions through the method of L- and TL-moments.

The main focus of this chapter is to model the extreme value data through Singh-

Maddala, Dagum and generalized Power function distribution. Second is to develop

the method of L- and TL-moments of these densities to estimate their parameters. To

avoid the problems those are associated with MoM and method of MLE in the

22

presence of heavy-tailed data. Specifically, the rest of the chapter is organized as

follows. In Section 3.2, Introduction of Singh-Maddala distribution and its parameter

estimation through the method of MLE, MoM and method of L- and TL-moments are

discussed. In Subsection 3.2.1 the L- and TL-moments of the Singh-Maddala

distribution are derived and moment ratios are given in Subsection 3.2.2. To compare

the estimation methods and highlight the properties of the estimates a comprehensive

simulation study is conducted in Subsection 3.2.3. A real data application is discussed

in Subsection 3.2.4 observing the monthly maximum temperature data of Jacobabad,

Pakistan.

The parameter estimation methods for Dagum distribution are developed and

compared in Section 3.3. The population L- and TL-moments are derived in the

Subsection 3.3.1, these moments are also used for parameter estimation. Subsection

3.3.2 is about the derivation of the moment ratios. To observe the precision and

accuracy of the methods of parameter estimation, a study based on Monte Carlo

simulation is provided in Subsection 3.3.3. In Subsection 3.3.4, a real data example is

also provided to compare the estimation of the method of L- and TL-moments with

the MoM and MLE procedure.

Parameter estimation of the generalized Power function distribution is discussed in

Section 3.4 with its graphical presentation and properties. In subsequent Subsections

3.4.1, 3.4.2 and 3.4.3 the C-moments, L-moments, TL-moments and their moment

ratios are derived for generalized Power function distribution respectively. In

Subsection 3.4.4 the method of MLE is also derived for this distribution. In

Subsection 3.4.5, the parameter estimators are formulated by comparing the

theoretical and sample moments for all three considered methods. Subsection 3.4.6

presents the parameter estimation of the generalized Power function distribution by

MoM, method of L- and TL-moments using Monte Carlo simulated data and the

performance of these methods are compared by biasness, root mean square error

(RMSE), moments ratios and percentage of relative bias (RB%). Furthermore, L- and

TL-moment ratio diagrams are also the part of this subsection. In a real application, an

annual maximum rainfall data set is considered in Subsection 3.4.7. Some concluding

remarks are given in Section 3.5.

23

3.2 Parameter estimation of Singh-Maddala distribution

Singh and Maddala (1976) introduced the Singh-Maddala (SM) distribution and it

appeared in Econometrica and soon after, it was frequently used for the analysis of the

income, wealth, consumption, expenditure and related data. McDonald and Ransom

(1979) compared the Lognormal, Beta, Gamma and SM distribution for family

income data for the year 1960 and 1969 to 1975 by three method of estimation and

found that the fitting of SM distribution better even than Beta distribution. McDonald

(1984) considered many three and four parameters probability distributions to model

the grouped income distribution data and concluded that the performance of the SM

distribution is best among all other distributions. Atoda, Suruga and Tachibanaki

(1988) concluded that the SM distribution is more favorable than other candidate

distributions for the Japanese income survey grouped data of 1975.

Henniger and Schmitz (1989) considered various distributions to model the United

Kingdom family expenditure data for the period 1968–1983, but none of them

accepted for the whole data set except SM and Fisk distributions. Brachmann and

Trede (1996) analyzed the German household income data for 1984 – 1993 by the

distributional approach and found the SM and generalized Beta-II distributions best to

model such data. Dastrup et al. (2007) studied the disposable income data and found

that in three parameter distributions, the best-fitted distributions are the SM and

Dagum distributions. Guessous et al. (2014) compared six probability distributions to

model travel time and validated the supremacy of the SM distribution in many

aspects. Brzezinski (2014) modeled the empirical impact factor distribution and

observed that the performance of SM distribution much better than the other models,

those were considered previously for this type of data. Sakulski et al. (2014)

quantified several statistical distributions for the analysis of rainfall such as Extreme

Value, Frechet, Log-normal, Log-logistic, Rice, SM and Rayleigh probability

distributions for summer, autumn, winter and spring seasons and finally they stated

that for all seasons, SM distribution fits quite well. Shao et al. (2004) proposed the

extended three-parameter Burr XII distribution to model the flood frequency data, and

method of MLE was investigated for parameter estimation of this distribution.

To estimate the parameters of SM distribution Singh and Maddala (1976) used

regression method, Shah and Gokhale (1993) considered the maximum product of

spacing and Stoppa (1995) derived its maximum likelihood. Herein, we have used the

24

method of MLE, MoM and method of L- and TL-moments to estimate the parameters

of the SM distribution.

Let be a random sample of size , follows the SM distribution then its pdf is given

by

⁄ (3.1)

where and are the shape parameters and is the scale parameter

.

The corresponding cdf is given by

⁄ (3.2)

and the th moment of the SM distribution is as follows

⁄ ⁄

(3.3)

3.2.1 L- and TL-moments for Singh-Maddala distribution

In this section, the population L- and TL-moment for the SM distribution are derived.

3.2.1.1 L-moment

The th L-moment for the SM distribution is derived using (3.1) and (3.2) in (2.5) and

obtained in the following form

∑ (

)

∫ 0 ( (

)

)

1

( (

)

)

For convenience substitute ⁄ in above expression, we get

∑ (

)

∫ ⁄ [ ]

By expanding [ ]

binomially, we obtain

∑ (

)

25

∑ (

)

∫ ⁄

After simplification using beta function, we get

∑ ∑ (

)

(

)

(

)

(3.4)

where is the beta type-II function defined by

∫

⁄ .

The first four L-moments are obtained by taking 1, 2, 3 and 4 in (3.4) as follows

0 ⁄ ⁄

1 (3.5)

(

) 0

⁄

⁄

1

(3.6)

(

) 0

⁄

⁄

⁄

1

(3.7)

(

) 0

⁄

⁄

⁄

⁄

1

(3.8)

Equating the population L-moments and corresponding sample L-moments, we can

estimate the parameter of the distribution.

3.2.1.2 TL-moments

The th TL-moment for SM distribution is derived using (2.11), (3.1) and (3.2) and

obtained in the following form

∑ (

)

∫ 0 ( (

)

)

1

( (

)

)

26

This expression is also simplified using Beta function as the th L-moment is solved

and obtained the following form

∑ ∑ (

)

(

)

(

)

(3.9)

The first four TL-moments are obtained by taking 1, 2, 3 and 4 in (3.9) as follows

(

) 0

⁄

⁄

1 (3.10)

(

) 0

⁄

⁄

⁄

1 (3.11)

(

) 0

⁄

⁄

⁄

⁄

1

(3.12)

(

) 0

⁄

⁄

⁄

⁄

⁄

1

(3.13)

Equating the population and corresponding sample TL-moments, we can estimate the

parameter of the SM distribution through simultaneous equation solution.

3.2.2 L- and TL-moments ratios

The L- and TL-moments ratios are analogues to the C-moments ratio and have the

same interpretation but L- and TL-moments summarize the probability distribution

more accurately than the conventional measures (Hosking, 1990). The first L-moment

ratio lies in the range . The is used to measure the asymmetry and it lies

between 0 and 1. Hosking and Wallis (1995) proved the range of the and it lies

between ⁄ and 1. The population L-moments ratios for SM distribution

are obtained as follows

27

0 ⁄

⁄ 1 (3.14)

⁄

⁄

⁄

⁄

⁄

(3.15)

⁄

⁄

⁄

⁄

⁄

⁄

(3.16)

The population TL-moments ratios are derived and obtained as follow

[

⁄

⁄

⁄

]

[ ⁄

⁄

] (3.17)

[ ⁄

⁄

⁄

]

0 ⁄

⁄

⁄

⁄

1

(3.18)

[ ⁄

⁄

⁄

]

0 ⁄

⁄

⁄

⁄

⁄

1

(3.19)

The general range of TL-moments is still not available in the literature.

3.2.3 Simulation Study

A simulations study has been carried out for two purposes. First, is to investigate and

compare the performance of the MLE, MoM, L- and TL-moments estimation

techniques. Second, is to explore the impact of sample size on estimation techniques.

28

Keeping it in mind, we present empirical analysis based on simulated data, to

compare the properties of the estimation methods for the SM distribution on their

bias, mean square error of estimates (MSE). The data is simulated using the R-

language assuming different sample sizes, (25, 50 and 200) and assuming

different value of each parameter and each sample is repeated 1000 times. In

simulation experiment each sample size is repeated to obtain the precision and

accuracy. The summary of the results for Maximum likelihood estimates (MLEs),

Method of moments estimates (MMEs), L-moments estimates (LMEs) and TL-

moments estimates (TLMEs) are presented in Table 3.1 and it is self-explanatory. In

general the TL-moments trim the outliner values from both sides of the data and the

data become less skewed, in this way its estimates are better than L-moments

estimates. The parameter estimates of L-moments are accurate and efficient than the

estimates attained using MLEs, particularly from small sample and approximately

equal in large sample. All three types of moment ratios also computed to summarize

the data.

Table 3.1: Summary of the average bias and MSEs of all the estimators ( ) of the

Singh-Maddala distribution for different sample size.

Bias ( )

Small ( ) ( )

Moderate

Large ( ) ( )

MSE( )

Small ( ) ( )

Moderate

Large ( ) ( )

3.2.4 Application

In this section, we have compared the four considered estimation method using

monthly maximum temperature data of Jacobabad, Pakistan. Jacobabad is one of the

hottest city of Pakistan, and its highest recorded temperature is 52.8 °C. The

geographical location of this city has Latitude 28.28 North and Longitude 68.45 South

29

and it is famous due to the consistently highest temperature in South Asia. The data of

the monthly maximum temperature of Jacobabad retrieved from the Pakistan

Meteorological Department (PMD) Islamabad. The length of data is 391 records from

January 1981 to December 2013 excluding an unobserved or unreported month.

In order to compare the estimation methods, the Kolmogorov-Smirnov ( ) goodness

of fit test is considered and fitting of SM distribution is also displayed graphically.

The smaller value of the -test, better the method will be. Where

1

1max ( ) , ( ) .i i

i n

i iKS F Y F Y

n n

The SM-distribution is a three-parameter distribution; therefore, the first three sample

moments are equated with the population moments to compute the parameter

estimates. Herein Newton-Raphson iterative estimation is employed for the solution

of the system of nonlinear equations and finally found out the estimates for all four

considered estimation methods. The results are reported in Table 3.2.

Table 3.2: Parameter estimates of the SM-distribution using different parameter

estimation methods and the result of the goodness of fit test.

Estimation Method Parameter Estimate -test p-value

TL-moments

5.17328

51.6708

5.93200

0.0700 0.0417

L-moments

5.48775

61.7679

17.1670

0.0853 0.0064

C-moments

6.07706

36.0512

1.49854

0.1430 1.94e-7

Maximum

Likelihood

5.46013

117.570

611.647

0.1085 0.0002

It has been observed that the method of TL-moments has provided better fitting and

estimates as compared to the other methods. As the -test has minimum value for

the method of TL-moments. The fitting of the SM distribution using estimates of the

considered methods are presented graphically as follows:

30

Figure 3.1: Empirical and fitted cdf of Singh-Maddala distribution using TL-

moments estimates

Figure 3.2: Empirical and fitted cdf of Singh-Maddala distribution using L-moments

estimates

31

Figure 3.3: Empirical and fitted cdf of Singh-Maddala distribution using method of

moments estimates

Figure 3.4: Empirical and fitted cdf of Singh-Maddala distribution using MLE

estimates

It is obvious from empirical results and Figure 3.1 to 3.4 that the method of TL-

moments has produced more accurate fitting on the real data set and the second good

fitted estimation method is the method of L-moments. Same conclusion is drawn from

the presentation of the PP-plots.

32

Finally, it is observed that the MoM does not produce satisfactory results. Method of

L- and TL-moments provide better results than the method of MLE and MoM.

Figure 3.5: PP-plots for all considered estimation methods for Singh-Maddala

distribution

According to the statistical and graphical presentations, TL-moments estimation

method is better as it has provided the superior fit on the data set.

The moments and moment ratios are commonly used to find the characteristics of the

probability distribution of the observed data set. These moments are calculated using

C-, L- and TL-moments and presented in Table 3.3.

Table 3.3: C-, L- and TL-moments and moments ratios for monthly maximum

temperature of Jacobabad

C-moments L-moments TL-moments

1st 34.168350 34.168350 34.411185

2nd

1219.9000 4.1591960 2.4469623

3rd

45186.230 -0.2428311 -0.1919081

4th

1724723.0 0.0809250 0.0596961

0.2121728 0.1217266 0.0711095

-0.2060238 -0.0583841 -0.0784271

33

3.3 Parameter estimation of Dagum distribution

Dagum distribution is extensively used for modeling a wide range of data in several

fields. It is a worthwhile option for analyzing income distribution, actuarial,

meteorological and equally preferable for survival analysis. It belongs to the

generalized Beta distribution and is generated from generalized Beta-II by considering

a shape parameter one and referred as inverse Burr distribution. Dagum (1977) and

Fattorini and Lemmi (1979) derived the Dagum distribution independently. Dagum

(1980) studied the income and income-related data by Dagum distribution. Dagum

(1983) also fitted this distribution on family income data for the United States of the

year 1978 and showed that its performance is the best among all the distributions.

Dagum (1990) found the superiority of the Dagum distribution in the income data

analysis of several countries. Bordley, McDonald, and Mantrala (1996) also studied

the United States family income data by probability distributions along with the

Dagum distribution. Botargues and Petrecolla (1997) assessed the income distribution

data of the Buenos Aires region and applied various types of Dagum distribution on it

to describe the data. Bandourian, McDonald and Turley (2003) revealed that the

Dagum distribution is the best among two and three parameter distributions by studied

the income data of the 23 countries. Quintano and Dagostino (2006) studied single-

person income distribution data of European countries and found that the Dagum

distribution performs better to model the each country data separately. Perez and

Alaiz (2011) analyzed the personal income data for Spain by Dagum distribution.

Alwan, Baharum and Hassan (2013) tried more than fifty distributions to model the

reliability of the electrical distribution system, and the Dagum distribution was

considered as the best choice. Herein, very few studies we have been cited but various

other related studies also confirm the better performance of the Dagum distribution.

Domma, Giordano and Zenga (2011) and Domma (2007) estimated the parameters of

Dagum distribution with censored samples and by the right-truncated Dagum

distribution respectively by maximum likelihood estimation. McGarvey, at al. (2002)

studied the estimation and skewness test for the Dagum distribution. Shahzad and

Asghar (2013) estimated the parameter of this distribution by TL-moments. Oluyede

and Rajasooriya (2013) introduced the Mc-Dagum distribution. Oluyede and Ye

(2013) presented the class of weighted Dagum and related distributions, and Domma

and Condino (2013) proposed the five parameter beta-Dagum distribution.

34

Let be a random sample of size then the pdf of the Dagum distribution is given by

⁄ (3.20)

where and are the shape parameters and is the scale parameter

.

The corresponding cdf is given by

⁄ (3.21)

and the th moment of the Dagum distribution is in the following form

⁄ ⁄

(3.22)

The quantile function of three parameter Dagum model is as

( ⁄ ) ⁄

(3.23)

where represent the cdf.

3.3.1 L-and TL-moments for Dagum distribution

In this section, the population L- and TL-moment for the Dagum distribution are

derived.

3.3.1.1 L-moment

The th L-moment for Dagum distribution is derived using (2.4), (3.21) and (3.23) in

the following form

∑ (

) (

)

∫ ⁄ ⁄ [ ]

For convenience substitute ⁄ in above expression, we get

∑ (

) (

)

∫ ⁄

After simplification using beta function, we get

∑ (

) (

)

(

) (3.24)

The first four L-moments are obtained by taking 1, 2, 3 and 4 in (3.24) as follows

35

0 ⁄ ⁄

1 (3.25)

(

) 0

⁄

⁄

1

(3.26)

(

) 0

⁄

⁄

⁄

1

(3.27)

(

) 0

⁄

⁄

⁄

⁄

1

(3.28)

Using population and corresponding sample L-moments of the Dagum distribution,

we can estimate the parameters of the distribution.

3.3.1.2 TL-moments

The th TL-moment for Dagum distribution is obtained using (2.11), (3.21) and

(3.23). Assuming Beta function and after simplification we get the following form

∑ ∑ (

)

(

)

(

)

(3.29)

The first four TL-moments are obtained by taking 1, 2, 3 and 4 in (3.29) as

follows

(

) 0

⁄

⁄

1 (3.30)

(

) 0

⁄

⁄

⁄

1 (3.31)

36

(

) 0

⁄

⁄

⁄

⁄

1

(3.32)

(

) 0

⁄

⁄

⁄

⁄

⁄

1

(3.33)

To estimate the parameters of the Dagum distribution by the method of TL-moments,

first three population and corresponding sample TL-moments are used.

3.3.2 L- and TL-moments ratios

The L- and TL-moments moments ratios for Dagum distribution are obtained using

the first four L- and TL-moment of the Dagum distribution. The population L-

moments ratios are derived as follows

0 ⁄

⁄ 1 (3.34)

⁄

⁄

⁄

⁄

⁄

(3.35)

⁄

⁄

⁄

⁄

⁄

⁄

(3.36)

The population TL-moments ratios are derived and obtained as follow

[

⁄

⁄

⁄

]

[ ⁄

⁄

] (3.37)

37

[ ⁄

⁄

⁄

]

0 ⁄

⁄

⁄

⁄

1

(3.38)

[ ⁄

⁄

⁄

]

0 ⁄

⁄

⁄

⁄

⁄

1

(3.39)

3.3.3 Simulation Study

A simulations study is carried out in order to investigate the performance of

estimation methods for the estimation of Dagum distribution and to study the effect of

the sample variability on estimation techniques. In this simulation study, the same

procedure is adopted as in Section 2.3.3. The four considered estimation techniques

are compared for Dagum distribution with respect to their bias and MSEs. The data is

simulated using the R-language assuming different sample sizes (25, 50 and 250)

and assuming different value of each parameter. In simulation experiment each

sample size is repeated to obtain the precision and accuracy. The summary of the

MLEs, MMEs, LMEs and TLMEs are presented in Table 3.4 and these results are

self-explanatory. The moment ratios are also computed to summarize the data.

Table 3.4: Summary of the average bias and MSEs of all the estimators ( ) of the

Dagum distribution for different sample size.

Bias ( )

Small

Moderate

Large ( )

MSE( )

Small

Moderate

Large ( )

38

3.3.4 Application

This application compared the four estimation methods, L- and TL-moments, MoM

and MLE to model the annual maximum wind speed at Vancouver, Canada for the

period of 1947-1984. This data set is given in the book of Reiss et al. (2007). The

descriptive statistics using moments and moment ratios for this data set are presented

in Table 3.5.

Table 3.5: C-, L- and TL-moments and moments ratios for annual maximum wind

speed at Vancouver

C-moments L-moments TL-moments

1st 64.92632 64.92632 64.286605

2nd

4309.149 5.494595 2.7119711

3rd

292476.7 0.639711 0.2484005

4th

20305540 0.974643 0.3664404

0.151110 0.084628 0.0421856

0.583328 0.116425 0.0915941

The estimated values of parameters for three-parametric Dagum distribution are

obtained by employing method of L- and TL-moments, MoM and method of MLE

methods using Newton-Raphson methodology. Table 3.6 contains the parameters

estimates and the tests for the goodness of fit criterion. On the basis of these results

method of TL-moments provided the most accurate results than the other considered

estimation methods.

Table 3.6: Parameter estimates of the Dagum distribution using different parameter

estimation methods and the result of the goodness of fit test.

Estimation

Method

Parameter

Estimate

KS-test

Statistic p-value

AD-test

Statistics p-value

TL-moments

10.2253

58.9903

1.84370

0.0824 0.9585 0.2598 0.9648

L-moments

11.0140

61.9682

1.27941

0.0884 0.9277 0.2386 0.9761

C-moments

6.77965

36.4534

20.5188

0.2607 0.01143 2.7461 0.0372

Maximum

Likelihood

6.38374

37.3924

24.5866

0.1657 0.2477 1.1883 0.2717

39

The following graphical presentations show the fitting of the Dagum distribution with

different estimation method on the annual maximum wind speed data at Vancouver,

Canada.

Figure 3.6: Empirical and fitted cdf of Dagum distribution using TL-moments

estimates

Figure 3.7: Empirical and fitted cdf of Dagum distribution using L-moments

estimates

40

Figure 3.8: Empirical and fitted cdf of Dagum distribution using MoM-moments

estimates

Figure 3.9: Empirical and fitted cdf of Dagum distribution using MLE estimates

41

Figure 3.10: PP-plots for all considered estimation methods for Dagum distribution

The results and graphical presentation have shown that the method of TL-moments is

the most accurate method, method of L-moments is the second best, while the third is

the method of MLE and MoM not being significant enough for parameter estimation.

3.4 Parameter estimation of generalized Power Function

distribution

The Generalized Power Function (GPF) distribution is a beta type distribution and it

is normally used for the analysis of heavy-tailed data. The GPF distribution has

diversity in shape and useful for extreme value frequency analysis. The parameter

estimation of this distribution with traditional estimation methods is either very

difficult or impossible. Therefore, its parameter estimation by the method of L- and

TL-moments will add some more valuable features in its application.

The GPF distribution is expressed in term of three parameters , where is the

shape parameter and , are the boundary parameters. Let random variable have

GPF distribution with the following form of the pdf

42

(

)

(3.40)

The corresponding cdf is of the form

(

)

(3.41)

The GPF distribution has the following properties

(i) Expectation,

,

(ii) Variance,

,

(iii) Reliability function, (

)

,

(iv) Hazard rate,

,

(v) Quantile function, ⁄ .

The two special cases of GPF distribution are discussed by Shahzad and Asghar

(2015), by considering , and , . Saran and Pandey (2004)

used the pdf given in (3.40) for the record value analysis. Ariyawansa and Templeto

(1986) made the statistical inference of three parameter Power function distribution

based on unordered sample. Considering two parameter GPF distribution,

Ahsanullah (1973) discussed its characteristics and Meniconi and Barry (1996) found

that it is a better option for reliability analysis. This distribution can assume various

shapes as graphed in Figure 3.11 and it could become a good candidate distribution

for survival analysis, record values and for flood frequency analysis.

43

Figure 3.11 The pdf’s of GPF distribution for various choice of parameters:

and [ ] ; , and [ ] ; ,

[ ] and ; 0, 10 and 2.0[2.0]10 with solid, dashed,

dotted, dotdash and longdash lines respectively.

3.4.1 Method of Moments and moments ratios of the GPF distribution

Most popular moments are the C-moments and these moments introduced by Karl

Pearson (1894). C-moments are used to find out the descriptive statistics and also

used for parameter estimation. This method is conceptually simple and easy, but

estimators are biased especially for small sample size and for the skewed data

(Sankarasubramanian & Srinivasan, 1999). First four C-moments for GPF distribution

are derived and reported as follows:

(3.42)

44

[

] (3.43)

0

{

}1 (3.44)

0

2

(

)31 (3.45)

To find the C-moments ratios such as coefficient of variation , skewness

and Kurtosis

, C- moments are converted to central moments and then

obtained by their respective formula. These ratios are dimensionless and are obtained

as follows

√

(3.46)

(3.47)

and

(3.48)

3.4.2 L-moments and L-moment ratios of the GPF distribution

The th L-moment for the GPF distribution is derived using (2.4), (3.40) and (3.41) in

the following form

∑ ∑ (

) ( )

0

1 (3.49)

The first four L-moments of the GPF distribution are given by

(3.50)

(3.51)

(3.52)

(3.53)

45

L-moments ratios such as , and for GPF distribution are derived and

obtained as follows

(3.54)

(3.55)

and

(3.56)

respectively.

3.4.3 TL-moments and TL-moment ratios of the GPF distribution

The th TL-moment for GPF distribution is obtained using (2.11), (3.40) and (3.41) in

the following form

∑ ∑ (

) (

)

0

1

(3.57)

The first four L-moments of the GPF distribution are given by

(3.58)

(3.59)

(3.60)

and

(3.61)

respectively.

TL-moments ratios such as , and for GPF distribution are derived and

obtained as follows

(3.62)

46

(3.63)

(3.64)

3.4.4 Parameter estimation of GPF Distribution

3.4.4.1 Maximum Likelihood Estimation of GPF distribution

Let be a sample of size from a GPF distribution then the log-likelihood

function for the GPF distribution has the following form

∑

(3.65)

In order to find the maximum likelihood estimators for , and , the partial

derivatives of (3.65) are taken with respect to the parameters to maximize the log-

likelihood function

∑

(3.65)

∑

(3.66)

(3.67)

But explicit solution from this system of equations is not possible theoretically and

numerically. This result is similar to the Hirano and Jack (2003) result as the MLE is

generally inefficient for the models those having parameter-dependent support. In the

same sense the MLE is not an appropriate technique to estimate the parameters of the

GPF distribution. Therefore, in this study, the method of MLE is not used further for

the parameter estimation of the GPF distribution.

3.4.4.2 Method of Moments Estimator

The GPF distribution has three parameters , and . Therefore, first three sampled

and theoretical C-moments are equated to derived the estimators. In this way, we

obtained three non-linear systems of equations and the simultaneous solution of these

equations did not provide close form estimators. So, herein parameters of GPF

distribution are estimated for MoM by Newton Raphson method.

47

3.4.4.3 L-Moments Estimator

L-moments estimators for the parameters of the GPF distribution are derived by

equating first three theoretical (3.50, 3.51, 3.52) and sampled L-moments. This

relationship gives us three equations and by solving these equations simultaneously,

the following estimators are obtained

(3.69)

(3.70)

(3.71)

3.4.4.4 TL-Moments Estimators

Using the same above stated methodology, TL-moments estimators are derived for the

parameters ( , , ) of GPF distribution and are given below

(3.72)

(

)

(

) (3.73)

and

(

)

(

)

(

)

(

)

(

)

(3.74)

By L- and TL-moments estimators for all unknown parameters are found in the close

form. This property expresses the superiority of these moments over the maximum

likelihood and MoM.

3.4.5 Comparison of L- and TL-Moments by Simulation Study

A simulation study is carried out to compare the properties of the L- and TL-moments

estimators for the GPF distribution. In this study different sample sizes {25, 50,

100, 200, 400, 800, 1500} for different parametric values such as {0.0, 10, 25, 50,

100}, {50, 100, 200, 500, 1000} and {0.5, 1.5, 2.5, 5.0, 7.0} are considered.

Taking all combinations of said parametric values, each sample is replicated 10,000

times to obtain the results. Many Matlab algorithms were coded for this simulation

study.

48

(a) 0.0, 10, 0.5 (b) 0.0, 10, 1.5 B

ias

(c) 0.0, 10, 2.5 (d) 0.0, 10, 5.0

Bia

s

(e) 0.0, 10, 7.0

Bia

s

Sample size

Figure 3.12: Bias of L- and TL-estimators by fitting the GPF distribution for

simulated data

0 200 400 600 800 1000 1200 1400 -0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 200 400 600 800 1000 1200 1400 -3

-2

-1

0

1

2

3

0 200 400 600 800 1000 1200 1400 -4

-3

-2

-1

0

1

2

0 200 400 600 800 1000 1200 1400 -6

-4

-2

0

2

4

6

8

0 200 400 600 800 1000 1200 1400 -30

-20

-10

0

10

20

30

LM- LM- LM- TLM- TLM- TLM-

49

(a) 0.0, 10, 0.5 (b) 0.0, 10, 1.5

RM

SE

(c) 0.0, 10, 2.5 (d) 0.0, 10, 5.0

R

MS

E

(e) 0.0, 10, 7.0

R

MS

E

Sample Size

Figure 3.13: RMSE of L- and TL-estimators by fitting the GPF distribution for

simulated data

0 200 400 600 800 1000 1200 1400 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 200 400 600 800 1000 1200 1400 0

5

10

15

20

25

0 200 400 600 800 1000 1200 1400 -50

0

50

100

150

200

250

0 200 400 600 800 1000 1200 1400 0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400 0

20

40

60

80

100

120

140

160

180

LM- LM- LM- TLM- TLM- TLM-

50

L-Moment TL-Moment

Sample Size

Figure 3.14: Bias and RMSE of L- and TL-estimator β for GPF distribution for

simulated data

The biases and RMSEs have been obtained by varying shape parameter , fixing the

value of boundary parameter and using L- and TL-moment

estimators. Magnitude of the biases is plotted in Figure 3.12 to observe the effect of

sample size and estimation method. Figure 3.12 shows that in all cases L-moments

estimators produce nearly unbiased estimates and as the sample size become large, it

produce nearly unbiased to truly unbiased results. TL-moments estimators produced

the comparatively biased and unstable results. The RMSE pattern is plotted in Figure

3.13. It is observed that the RMSEs of L-moments estimators are minimum as

compare to TL-moments estimators. It is also noted that there is a positive

relationship between the RMSE and the value of parameters, as we increase the value

of shape parameter the RMSE also increases as Figure 3.13 shows.

0 200 400 600 800 1000 1200 1400-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Bia

s

=50

=100

=200

=500

=1000

0 200 400 600 800 1000 1200 1400-0.5

0

0.5

1

1.5

2

2.5

3

3.5

Bia

s

=50

=100

=200

=500

=1000

0 200 400 600 800 1000 1200 14000

2

4

6

8

10

12

14

16

Ro

ot M

ea

n S

qu

are

Err

or

=50

=100

=200

=500

=1000

0 200 400 600 800 1000 1200 14000

5

10

15

20

25

Ro

ot M

ea

n S

qu

are

Err

or

=50

=100

=200

=500

=1000

51

In Figure 3.14, the biases and RMSEs are presented for the various values of upper

bound location parameter using L- and TL- moments estimators. Both types of

estimators produce small amount of bias, as in the case of 1000 the L- and TL-

moments estimators gives 1.4120 and 3.1873 bias respectively. But comparatively L-

moments estimator for is more precise and accurate in the sense of bias and RMSE.

Same procedure is repeated for the lower bound location parameter , in this case

again L-moment estimator produced less biases and RMSEs as compared to its

counterpart TL-moment estimator. Over all L-moments estimators are more precise

than other estimators.

Figure 3.15: L-moment ratio diagram for

and curve

Figure 3.16: TL-moment ratio diagram for and

curve

and are often used to summarize the shape of the distribution (Hosking, 1992),

and for this 200 independent random samples have been generated to estimate the

-0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.050

0.05

0.1

0.15

0.2

0.25

L-K

urt

osis

L-Skewness

-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

TL

-Ku

rto

sis

TL-Skewness

52

coefficients. L- and TL-moment ratio diagrams are presented using simulation results

in Figure 3.15 and Figure 3.16. It is observed that there is inverse relationship

between L-Skewness ( ) and L-kurtosis (

) and similarly in TL-Skewness

( ) and TL-kurtosis (

).

Table 3.7: and for three type of moments assuming varying parametric

values and for 25.

True Parameters RB%

L-Moment Estimates

0.0 10 0.5 0.5080 0.2030 0.0038 30.1358

RB

% f

or

1.5 0.2529 -0.0924 0.0284 3.0483

2.5 0.1682 -0.1770 0.0674 1.0517

5.0 0.0916 -0.2475 0.1072 0.3479

7.0 0.0670 -0.2691 0.1223 0.1470

0.0 50 5.0 0.0920 -0.2476 0.1091 0.1411

RB

% f

or

100 0.0917 -0.2475 0.1081 0.1439

200 0.0914 -0.2473 0.1087 0.1603

500 0.0918 -0.2475 0.1094 0.1269

1000 0.0916 -0.2471 0.1084 0.1454

1.0 25 2.5 0.1586 -0.1764 0.0674 -148.6

RB

% f

or

2.5 0.1452 -0.1764 0.0676 99.645

5.0 0.1240 -0.1760 0.0689 11.647

7.0 0.1083 -0.1763 0.0679 25.75

10 0.0866 -0.1766 0.0682 45.077

TL-Moment Estimates

0.0 10 0.5 0.3433 0.1627 0.0062 32.9733

RB

% f

or

1.5 0.1451 -0.0668 0.0191 187.346

2.5 0.0919 -0.125 0.0407 -12.4614

5.0 0.0481 -0.1705 0.0595 -88.0517

7.0 0.0347 -0.1848 0.0681 -98.2526

0.0 50 5.0 0.0482 -0.1703 0.0618 0.2844

RB

% f

or

100 0.0481 -0.1701 0.0603 0.2954

200

0.0479 -0.1695 0.0610 0.3290

500

0.0480 -0.1702 0.0627 0.2606

1000

0.0480 -0.1696 0.0606 0.3093

1.0 25 2.5 0.0867 -0.1246 0.0414 -1000

RB

% f

or

2.5

0.0796 -0.1241 0.0405 -645.9

5.0 0.0681 -0.1233 0.0420 130.4

7.0 0.0597 -0.1236 0.0405 176.28

10 0.0479 -0.1238 0.0479 77.083

53

Table 3.8: and for three type of moments assuming varying parametric

values and for 1500.

True Parameters RB%

L-Moment Estimates

0.0 10 0.5 0.5002 0.2002 0.0001 0.2738

RB

% f

or

1.5 0.2500 -0.0909 0.0260 0.0225

2.5 0.1668 -0.1760 0.0643 0.0064

5.0 0.0909 -0.2499 0.1071 0.0034

7.0 0.0667 -0.2727 0.1224 -0.0006

0.0 50 5.0 0.0909 -0.2500 0.1072 0.0010

RB

% f

or

100 0.0909 -0.2501 0.1073 -0.0030

200 0.0909 -0.2500 0.1072 -0.0002

500 0.0909 -0.2498 0.1072 0.0038

1000 0.0910 -0.2501 0.1072 0.0007

5.0 250 5.0 0.0887 -0.2498 0.1071 -109.7

RB

% f

or

10 0.0889 -0.2491 0.1073 -50.09

25 0.0909 -0.2498 0.1071 -77.67

50 0.0912 -0.2508 0.1072 -82.48

100 0.0909 -0.2500 0.1073 -85.49

TL-Moment Estimates

.0 10 0.5 0.3336 0.1589 0.0001 0.0699

RB

% f

or

1.5 0.1429 -0.0653 0.0148 100.488

2.5 0.0910 -0.123 0.0349 1.2115

5.0 0.0476 -0.1709 0.0559 4.8281

7.0 0.0345 -0.1852 0.0631 11.5582

0.0 50 5.0 0.0476 -0.1710 0.0559 0.0025

RB

% f

or

100 0.0476 -0.1710 0.0559 -0.0029

200

0.0476 -0.1709 0.0559 0.0011

500

0.0476 -0.1707 0.0559 0.0068

1000

0.0476 -0.1712 0.0559 0.0007

5.0 250 5.0 0.0476 -0.1709 0.0559 -280.132 R

B%

for

10

0.0476 -0.1710 0.0559 -236.55

25 0.0476 -0.1712 0.0559 -53.92

50 0.0476 -0.1705 0.0559 -61.32

100 0.0476 -0.1709 0.0559 -82.29

All three L- and TL-moments ratios are computed and presented in Table 3.7 and

Table 3.8 for comparison. The is a positive relative measure of dispersion and it

lies between . It is noted that all moments ratios varies, with change in

value of the shape parameter. Nevertheless, there is no significant variation noted in

the estimates of the skewness and kurtosis due to the variation in other two parameters

. The reason may be that only is involved in expressions (3.55), (3.56), (3.63)

and (3.64). RB% of L- and TL-moments estimators in Table 3.7 and Table 3.8

provides evidence that L-moments estimators are more precise.

54

3.4.6 Application

To investigate the estimation and fitting of the MoM, method of L- and TL-moments,

annual maximum precipitation data of the Karachi city, Pakistan is considered. The

data consist of 59 annual maximum precipitation records for the years 1950–2009,

one of the value for the year 1987 is missing, rest of the recorded values are used for

the analysis. The MoM, method of L- and TL-moments estimates and their

coefficients for this data set are given in the following table.

Table 3.9: Parameter estimation using annual maximum precipitation data of Karachi,

Pakistan

Method of Estimation Estimates Coefficients

Method of Moments

32.7882

362.442

0.33532

CV = 0.8003

Sk = 1.0670

Kr = 3.8154

L-moments

20.2038

333.727

0.44463

CV = 0.4377

Sk = 0.2379

Kr = 0.0948

TL-moments

20.8355

297.663

0.50445

CV = 0.2653

Sk = 0.1563

Kr = 0.0339

Figure 3.17: Empirical, fitted cdf of Power Function distribution using TL-moments

estimates

55

Figure 3.18: Empirical, fitted cdf of Power Function distribution using L-moments

estimates

Figure 3.19: Empirical, fitted cdf of Power Function distribution using MoM

estimates

The comparison of MoM, method of L- and TL-moments is presented in Figure 3.17,

Figure 3.18 and Figure 3.19. This graphical presentation shows that the L-moments

provide better results and fitting than MoM and TL-moments for the GPF distribution.

56

3.5 Conclusion

It is a standard statistical practice to summarize a probability distribution or an

observed data set by some of its moments. Therefore, the L- and TL-moments for the

SM, Dagum, and the GPF distribution are derived to summarize the data accurately.

These derived moments are also used to estimate the parameters of these distributions

and then method of L- and TL-moments are established by equating the population

and sample moments. To compare these estimation methods with MLE and MoM, a

simulation study has been carried out for each three distributions. The results from

simulation study are indicated that the estimates of the L- or TL-moments are the least

bias with minimum MSE than the other considered estimation methods. It has become

more obvious as the sample size increased. These conclusions are also justified in real

datasets applications. Additionally, the shape of the distributions has been

investigated through L- and TL-moments numerically.

57

4. CHAPTER 4

Extreme Value Analysis by Transmuted Distributions

4.1 Introduction

The quality of statistical analysis is highly dependent on the assumed probability

distribution, and there exist a vast amount of literature on the choice of the best

probability distribution. It is very difficult to determine analytically the most suitable

distribution that universally acceptable. Therefore, one of the most interesting

research directions for mathematicians and statisticians is to develop an appropriate

class of distributions along with their relevant statistical properties and

methodologies. The goal of this substantial effort is to design the standard probability

distributions for real world situations that serve as actual models. However, there are

still many important symmetric and asymmetric aspects present where the existing

distributions do not model the real data in a true sense. In particular, extreme value

frequency data pattern, because most of the models would not have suitable

probability density functions for studying the real phenomenon.

Keeping this in mind, another objective of this study is to investigate some probability

distributions to generalize them for analyzing and modeling real situation data,

especially extreme value frequency data. In this context, we selected Shaw and

Buckley (2009) proposed the QRTM, which applies to both symmetric and

asymmetric distributions as well as having other nice properties. The concept of the

QRTM defined in detail in section 2.4 as it provides a new generalization of any

distribution with an additional parameter. Therefore, the new generalized density

becomes more flexible than the parent density with a variety of shapes of survival and

hazard function, when modeling even the complex data sets of hydrological,

meteorological, engineering, basic, social and behavioral sciences.

58

This recent QRTM is used to obtain the more flexible and versatile model. Sharma et

al. (2014) proposed the transmuted inverse Rayleigh distribution and used this density

in survival analysis because its hazard function has the upside-down bathtub shaped.

Khan and King (2014) proposed a generalized transmuted inverse Weibull

distribution and found that it is better than the parent distribution in real data

application. Similar interpretations are observed by Ahmad et al. (2014), Khan at al.

(2014), Aryal (2013), Merovci (2013), Elbatal (2013) and Aryal and Tsokos (2011).

Rest of the Chapter organized as follows, in Section 4.2, the four parameter

transmuted Singh-Maddala distribution is introduced along with its properties and

application. In Subsection 4.2.1, moment, moment generating function and

random number generating process for transmuted Singh Maddala distribution are

derived. Survival analysis of the distribution such as reliability function, hazard rate

function is obtained and graphically presented in Subsection 4.2.2. In Subsection

4.2.3, order statistics and the densities of lowest, highest and joint order statistics are

specified. Generalized TL-moments and its special cases are derived in Subsection

4.2.4. Methodology for parameter estimation discussed in Subsection 4.2.5. The real

data set application of the transmuted Singh-Maddala distribution is presented in

Subsection 4.2.6.

In Section 4.3, the transmuted Dagum distribution is proposed with the mathematical

properties and application. In Subsection 4.3.1, moment and moment generating

function are derived and reported. Subsection 4.3.2 is about the quantile function,

median and random number generating process for transmuted Dagum distribution.

Properties of the reliability analysis and their graphical presentation are given in

Subsection 4.3.3. Subsection 4.3.4 is related to order statistics: the lowest, highest and

joint order densities of transmuted Dagum distribution are specified. Subsection 4.3.5

contains the generalized TL-moments and its special cases, such as L-, TL-, LL- and

LH-moments. Methodology for parameter estimation is discussed in Subsection 4.3.6.

To compare the suitability of transmuted Dagum distribution with its parent

distribution, rainfall data is selected, and its goodness of fit is measured with popular

criteria in section 4.3.7.

Section 4.4 is about the development of the transmuted New distribution with its

various properties and application. Reliability analysis of this distribution is obtained

in this Subsection 4.4.1. In Subsection 4.4.2 basic statistical properties of the

59

transmuted New distribution are studied through raw moments and random number

generation strategy is also discussed. The order statistics of transmuted New

distribution is provided in Subsection 4.4.3. To estimate the parameters, in Subsection

4.4.4 the maximum likelihood has been discussed for proposed distribution. In

Subsection 4.4.5 the empirical study is carried out using two real examples and found

that the transmuted New distribution have more advantageous than the baseline

distribution. Finally, in Section 4.5, we make some concluding remarks.

4.2 Transmuted Singh-Maddala distribution

The Singh-Maddala is a well-known distribution and there is wide monographic and

periodical literature available on it. This distribution attributes to Singh and Maddala

(1976), and its brief introduction has already been provided in Section 3.2. It was

initially derived for income data analysis but, later on, it widely used in actuarial,

economic, extremes and reliability studies. Zimmer et al. (1998) studied this model

and concluded that the model is good for failure time data analysis. In the study of

extremes, Shao et al. (2013) applied the extended Singh-Maddala (SM) distribution

for the flood frequency analysis. It performed better than its comparative models in

many fields. To enhance its applicability in various other fields, we introduced the

transmuted Singh-Maddala distribution in this section. The transmuted Singh-

Maddala (TSM) is more versatile and flexible than the SM distribution.

The TSM distribution is proposed using the QRTM taking the SM distribution as a

parent distribution. As provided in Section 3.2, let be a random sample of size

and come from the SM distribution with the pdf of the form

⁄

its cdf is as

⁄

Where and are the shape parameters and is the scale parameter

. The cdf and pdf of the TSM distribution are derived using cdf and pdf of the SM

distribution in (2.16) and (2.17) in the following form

[ ⁄ ][ ⁄ ] (4.1)

and

60

[ ⁄ ]

⁄ (4.2)

respectively.

The density function is sketched in Figure 4.1 for various

combinations of all the four parameters. It can be observed that the TSM density

shows the several behaviors those the SM density cannot attain.

Figure 4.1: The pdf’s of TSM distribution for various choice of parameters:

0.8[0.2]0.63; 2[1]7; 1.5[0.5]3.5; 3.0[0.5]5.0 and

1.5[0.5]3.5 with solid, dashed, dotted, dotdash and longdash lines respectively.

4.2.1 Basic Properties

In this Section, the main statistical properties for the TSM random variable are

derived.

3 [ ] , it means the range of the values is from to with increment of .

61

Figure 4.2: The cdf’s of TSM distribution for various choice of parameters:

0.8[0.2]0.6; 2[1]7; 1.5[0.5]3.5; 3.0[0.5]5.0 and


Theorem 4.1. Let the random variable follow TSM distribution, then its C-

moment is given by

⁄ 0 ⁄

⁄

1 (4.3)

Proof. By the definition of the C-moment of TSM distribution is given by

∫

∫

⁄ [ ⁄ ]

∫

⁄

∫

⁄

62

For convenience substitute ⁄ in above expression and taking the simple

steps, is obtained in the following form

⁄ ⁄ ⁄ ⁄ ,


∫

⁄ .

Taking the simple steps, we get the required result.

The mean of the TSM distribution is obtained by taking in (4.3) in the

following form

⁄ 0 ⁄

⁄

1 (4.4)

and variance is of this distribution derived in the following form

⁄ 0 ⁄

⁄

1

[ ⁄ ] 0 ⁄

⁄

1

(4.5)

respectively.

The moment ratios such as , and can be obtained by assuming

in (4.3) and using the usual formulas.

Theorem 4.2. The moment generating function of , , when random variable

follows TSM distribution is given by

∑

(4.6)

Proof. Let the moment generating function of be given by

∫

∫ .

/

∑

⁄ 0 ⁄

⁄

1

63

∑

The last expression is the required result. Quantile function

The random variable follows the cdf given in (4.1). The quantile function, say

is the inverse of the equation ( ) ,

[ ⁄ ][ ⁄ ]

Now simplifying it for , we get

[.

√ /

]

⁄

To obtain the quantiles (quartiles, decile and percentile) of the TSM distribution

simply replace with the desired value. The median is a specific form of the above

expression and the median of the TSM distribution is obtained as

[( √ ) ⁄

] ⁄

4.2.1.1 Random data generation

One can generate random data from distribution function of the TSM distribution

using the inverse transformation method, as

[ ⁄ ][ ⁄ ]

This yields

[.

√ /

]

⁄

(4.7)

where is standard uniform variate.

The in (4.7) follows TSM distribution and can be readily used to generate the

random data taking suitable values of the parameters , and .

4.2.2 Reliability analysis

In life data analysis the most common functions, reliability function and hazard

function are used to describe the life of a component or system. The role of the

reliability in extreme value data is also obvious. As the accurate reliability of

64

electronic or non-electronic components and systems are required in order to meet the

demands, safety and warranties of the products. Applications of extreme value theory

in reliability analysis can be found in many fields but we refer the thesis of the Kuhla

(1967) for more detail. So the reliability and hazard functions are discussed in this

Section.

4.2.2.1 Reliability function

The reliability function provides the probability of an item that is functioning for

a specific quantity of time without failure. The reliability function and cumulative

distribution function are reverse of each other. As and represent the

probability of survival and failure respectively. The reliability function of the TSM

distribution is given by

( (

)

)

0 ( (

)

)

1. (4.8)

4.2.2.2 Hazard function

Hazard function is the ratio of pdf and the reliability function. Hazard rate is an

important property of a random variable from survival analysis. It is used to find the

conditional probability of failure, given that it has survived at time . The hazard rate

for the TSM distribution is given by

(

)

0 ( (

)

)

1

( (

)

) 0 ( (

)

)

1

(4.9)

It can be observe that when , the behaviour of the hazard function is decreasing

and then constant. When , the behaviour of the hazard function is upside-down

bathtub shaped (increasing to maximum and then decreasing). Thus the TSM

distribution shows decreasing, increasing or unimodal hazard rate in specified ranges

of the parameter values. The various shapes of hazard function are presented in Figure

4.3 and Figure 4.4 assuming different combinations of parametric values.

65

Figure 4.3: The hazard functions of TSM distribution when 1.

Figure 4.4: The hazard functions of TSM distribution when 2.

Many survival studies eventually necessitated the hazard functions that instantly

increased to a maximum at the beginning of life and then gradually decreased until

stabilize.

66

4.2.3 Order statistics of the transmuted Singh-Maddala distribution

Order statistics of a random variable that satisfies the condition of ordering

, independently identically distributed, having a great interest in the

analysis of the extreme (smallest and largest), median and joint order statistics.

Usually interest lies in the lowest temperature in winter, median income distribution

in a country, highest flood flow in dams and joint breaking strength. We also derived

the density of the order statistics.

The density of the order statistics is defined by Arnold et al. (1992) and is given

by

( ) [ ( )]

[ ( )]

( ) (4.10)

Where [ ] .

The probability density of order statistics for TSM distribution is obtained by

substituting (4.2) and (4.1) in (4.10) and obtained as follow

( )

∑ ∑ ∑ (

) (

) (

)

(4.11)

( ( ⁄ ) )

[ ( ( ⁄ ) )

]

The density of the smallest order statistic, has the following form

( )

∑ (

)

( ( ⁄ )

)

[ ( ( ⁄ ) )

]

The density of the order statistic, is obtained from (4.11) in the following

form

( )

∑ ∑ (

) (

)

( ( ⁄ ) )

[ ( ( ⁄ )

)

]

67

The joint pdf of and for the TSM distribution is derived by

using the general expression given by Balakrishnan and Cohen (1991) is as given

below

∑ ∑ ∑ ∑ ∑ ∑

(

) (

) (

) (

) (

)

(

) ⁄

⁄ [ ( ( ⁄ ) )

].

4.2.4 Generalized TL-moments

TL-moments are a worthwhile contribution to extreme values analysis. These

moments, based on the order statistics and these moments describe the shape of the

probability distribution in a better way than C-moments. Elamir and Seheult (2003)

introduced and defined the th generalized TL-moment with smallest and largest

trimming as follows

∑ (

)

(4.12)

Accordingly the generalized TL-moment for TSM distribution has the following form

∑ ∑ ∑ (

) (

)

(

)

⁄

0 ⁄

⁄

1

(4.13)

Proof. The well-known density of the order statistics for is as

( ) [ ( )]

[ ( )]

( )

(4.14)

Where F and are the cdf and pdf of the TSM distribution, respectively. Now

substituting the cdf and pdf of TSM distribution given in (4.1) and (4.2) in (4.14). We

get

68

( )

∑ ∑ (

)

(

)

{ ( ⁄ ) }

[ ( ( ⁄ ) )

] (

)

(4.15)

Now using (4.15), the is obtained as follows

∑ ∑ (

)

(

) ⁄ 0

⁄

⁄

1

Finally substitute the in (4.12) and after taking the simple steps, we

obtained the generalized TL-moment for TSM distribution (

) in the form given

in (4.13).

4.2.5 Special cases of generalized TL-moment

The L-moments, TL-moments, LL-moments and LH-moments are introduced by

different authors independently. In 2003, Elamir and Seheult derived the generalized

form and these moments became the special cases of the generalized TL-moment.

4.2.5.1 TL-moments

Generally, it is possible to trim any number of smallest and largest values from the

ordered observation. As a special case only one extreme value from both sides are

trimmed then the th TL-moment with is derived as

∑ ∑ ∑ (

) (

)

(

)

⁄ 0

⁄

⁄

1

69

To derive the first four TL-moments substitute .

4.2.5.2 L-moments

When none of the observation is trimmed from the ordered sample,

generalized TL-moment reduced to L-moments and basically L-moments and related

moments are due to the Hosking (1990) methodology. The th L-moment of TSM

distribution is obtained as follows

∑ ∑ ∑ (

) (

)

(

)

⁄

0 ⁄

⁄

1

4.2.5.3 LL-moments

LL-moments progressively reflect the characteristics of the lower part of the

distribution. Bayazit and Onoz (2002) introduced these moments and later it became

the special case of generalized TL-moment, when and . Following is the

th LL-moment

∑ ∑ ∑ (

) (

)

(

)

⁄ 0

⁄

⁄

1

4.2.5.4 LH-moments

LH-moments proposed by Wang (1997), these moments describe the upper part of the

data more precisely. These moments give more weight to the larger values and the

theoretical LH-moments for the TSM distribution are derived as

∑ ∑ ∑ (

) (

)

(

)

70

⁄

0 ⁄

⁄

1

The LH-moments for can be evaluated but Wang (1997) preferable

suggested the value of up to four only.

4.2.6 Parameter estimation

In this section, the interest is to estimate the parameters of TSM distribution by the

method of MLE.

Let be independently distributed random variables of size . Then the

sample likelihood function for this distribution is given as

(

)

∏ ⁄

[ ⁄ ]

The sample log-likelihood function corresponding to the above expression is obtained

as

∑ ⁄

(4.16)

∑

∑ [ ⁄ ]

Taking the first order derivatives of (4.16) with respect to the parameters and equating

the resulting expressions equal to zero to find the maximum likelihood estimators, we

obtain the following equations

∑

∑ ⁄ ⁄

⁄

∑ ⁄ ⁄ ⁄

⁄

(4.17)

∑

⁄

⁄

∑ ⁄ ⁄

[ ⁄ ]

(4.18)

71

∑ ⁄

∑ ⁄ ⁄

[ ⁄ ]

(4.19)

∑ ⁄

⁄

(4.20)

The exact closed forms of maximum likelihood estimators are not possible, so the

estimates and of parameters and respectively are obtained by

solving the above four nonlinear equations analytically. The solution of the nonlinear

system of equations is conveniently possible by Newton Raphson algorithm. The two

sided confidence intervals of the estimates are also obtained using large sample

approximation properties.

4.2.7 Application

Household expenditure data is a good tool to measure the living standards and

consumption patterns in a society. This univariate datasets can properly describe by

probabilistic approach. The best fit distribution provides reliable knowledge about

data patterns to make policy in order to lead society to the direction of development,

especially for low income countries like Pakistan. In this study we used the monthly

household expenditure data from the Household Integrated Economic Survey (HIES)

for 2010-2011, this survey is conducted after every two years by Pakistan Bureau of

Statistics. HIES is the largest survey and best available source for the expenditure

data in Pakistan.

Table 4.1: Summary Statistics of expenditure data, HIES, Pakistan

Sample Size 15510

Minimum expenditure 325

Maximum expenditure 93300

0.20 Percentile 4117.0




Median 5969

Mean 6845

Standard deviation 4120.234

To compare the two distributions, we consider criteria AIC (Akaike information

criterion), AICC (corrected Akaike information criterion) and BIC (Bayesian

72

information criterion) for the data set. The best-fitted distribution produces smaller

values of the AIC, AICC, and BIC.

Table 4.2: Estimated parameters of TSM and SM distribution by MLE

Model Parameter Estimate AIC AICC BIC

Transmuted

Singh-Maddala

3.9412

5535.8248

0.8513

0.5175

293272.6 293272.6 293303.2

Singh-Maddala

3.2104

6262.0926

0.7723

293757.4 293757.4 293780.3

The variance-covariance matrix of the MLEs under the TSM distribution is computed

as

1

0.00269 3.11107 0.00108 0.00030

3.11107 31447.80291 1.59301 18.18336.

0.00108 1.59301 0.00059 0.00008

0.00030 18.18336 0.00008 0.01209

F

Thus, the standard deviation of the MLE for the and are

0.51936, ( ) 177.33528, ( ) 0.02436, ( ) 0.10997 respectively.

Therefore, 95% confidence intervals for the and are [3.83943, 4.04302],

[5188.248, 5883.402], [0.80352, 0.89902] and [0.30195, 0.73304] respectively.

The MLE parameter estimates, AIC, AICC and BIC corresponding to the fitted

models for the expenditure data set are presented in Table 4.2. According to these

results, the TSM distribution provides a better fit than the parent distribution.

Additionally, to test the significance of the transmuted parameter, likelihood ratio

(LR) test has been employed. To perform this test the maximized restricted and

unrestricted log-likelihoods can be computed under the following null and alternative

hypothesis

(restricted, SM model is true for the data set)

versus

(unrestricted, TSM model is true for the data set).

The LR-test statistic for testing the hypothesis is as follows

[ ( ) ( )],

73

Figure 4.5: Estimated and fitted densities on the empirical histogram for the data

set of household expenditures

Figure 4.6: Empirical, fitted TSM and SM cdf on the data set of household

expenditures

where and are the maximum likelihood estimates under and respectively.

The LR statistic is asymptotically distributed as chi-square For this data set,

is obtained and . Therefore the test statistic does not

74

support the null hypothesis and leads us to conclude, TSM distribution has provided

the better fitting model rather than the SM distribution.

The density plot and cdf plot compares the fitted densities with the empirical

histogram and cdf and are given in Figure 4.5 and Figure 4.6 respectively for the

observed expenditure data set. These plots also show that the TSM model is closer to

the empirical data than the SM Maddala model.

Table 4.3: First four L-, TL-, LL- and LH-moments and moments ratios for the

household expenditure data

L-moments TL-moments LL-moments LH-moments

1st 5781.51 5364.57 4286.17 7276.85

2nd

1495.34 677.626 808.807 1434.21

3rd

416.942 100.398 33.9845 521.938

4th

365.965 74.6485 99.9120 357.544

0.25864 0.12631 0.18875 0.19709

0.27882 0.14816 0.04201 0.36391

The first four moments and moment ratio are presented in Table 4.3. It is noticed that

the mean (1st moment) is the highest in LH-moments and lowest in LL-moments case,

the reason is that the LH-moments and LL-moments are introduced to present the

high and low part of data respectively. The variation (2nd

moment) in the case of TL-

moments and LL-moments is lowest and highest respectively because TL-moment

trimmed the extreme values of the data but L-moments based on the full data. In the

same way we can interpret the value of the . It is also observed that the and

are high in case of LH-moments and it is due to the trimming the lower value from the

data.

4.3 Transmuted Dagum distribution

Dagum (1977) introduced a new income distribution just after one year of SM

distribution and initially it was less popular than SM distribution. However, in recent

years, the Dagum distribution is a strong competitor of the SM distribution and a

more appropriate choice in many applications. Moreover, it is equally preferable for

actuarial, meteorological and survival studies along with the income data. The brief

detail of this distribution is provided in Section 3.3.

In this Section, we introduced an extended Dagum distribution, named as transmuted

Dagum distribution. The main motivation for generalizing a standard distribution is to

75

provide a flexible distribution to model a variety of data more perfectly. The extended

distribution has been expressed using the QRTM.

Let a random variable follows the Dagum distribution and having pdf and cdf given

in (3.20) and (3.21). Using these functions and the QRTM we derived the cdf of the

transmuted Dagum (TD) distribution and obtained as follows

⁄ [ ⁄ ] (4.21)

and its corresponding transmuted pdf is

[ ⁄ ]

⁄ (4.22)

respectively.

Figure 4.7: The pdf’s of TD distribution for various choice of parameters:

3.0[1.0]7.0, 2.0[0.5]4.0, 0.5[1.0]4.5 and 1.0[0.5]1.0

with solid, dashed, dotted, dotdash and longdash lines respectively.

76

Figure 4.8: The cdf’s of TD distribution for various choice of parameters:

3.0[1.0]7.0, 2.0[0.5]4.0, 0.5[1.0]4.5 and 1.0[0.5]1.0

with solid, dashed, dotted, dotdash and longdash lines respectively.

Different shapes of the transmuted density and distribution functions assuming

various combinations of parameters are illustrated in Figure 4.7 and Figure 4.8.

One can observe that the pdf and cdf of the TD distribution are more flexible with

variety of shapes than the parent Dagum distribution.

4.3.1 Basic properties

In this section, main statistical properties such as moments, mean, variance, and

moment generating function for TD distribution are derived and discussed.

Theorem 4.3: Let the random variable follows the TD distribution, then its

moment has the following form

(

) 0

⁄

⁄

1 (4.23)

77

Proof By definition, the C-moment for TD distribution is given by

∫ [ ⁄

⁄ ]

for convenience we substituted ⁄ , hence

∫ ⁄

∫ ⁄

= * (

) (

)+,


∫

by simplification, we obtained the result given in (4.23).

In particular, by setting and in (4.23), we obtain mean and variance in the

following form

(

) 0

⁄

⁄

1

and

(

) [ ] [ (

)]

[ ]

respectively, where (

) ⁄

To obtain the expression for the C-moment ratios such as , and , the

with its specific values are used.

Theorem 4.4: The moment generating function of , , when random variable

follows TD distribution is

∑ ⁄

[ ] (4.24)

Proof. Let the moment generating function of is given by

78

∫

∫ .

/

∑

∫

The last expression is the required result.

4.3.2 Quantile function and random data generation

Quantile function divides the ordered data into equal size portions. Hyndman and

Fan (1996) defined the quantile function is as follow

{ } (4.25)

where is the distribution function.

According to this definition, the quantile ( )Q q for TD distribution is obtained as

follows

[( √

)

⁄

]

⁄

(4.26)

The median (second quartile) of the TD distribution is given by

[( √ ) ⁄

] ⁄

.

And let suppose the is the standard uniform variate in (4.26) instead of . Then the

random variable

[( √

)

⁄

]

⁄

follows the TD distribution assuming parameters , , and known.

4.3.3 Properties of TD distribution in term of reliability analysis

In this section, we derive and describe the behavior of the reliability and hazard rate

function of the TD distribution.

79

4.3.3.1 Reliability function

The reliability function, gives the probability of surviving of an item at least

reach the age of time . The reliability function of a TD distribution is given by

∫

⁄ [ ⁄ ]

The pattern of the reliability function for TD distribution is sketched in Figure 4.9 for

various combinations of parametric values.

Figure 4.9: The various shapes of reliability function for TD distribution

4.3.3.2 Hazard rate function

One of the most important property of the reliability analysis is the hazard rate

function. It measures the inclination towards failure rate. The probability approaches

failure increases as the value of the hazard rate increase. Mathematically, the hazard

rate function of TD distribution is derived as follows

0 ( (

)

)

1

( (

)

)

[ ( (

)

)

0 ( (

)

)

1]

(4.27)

80

The hazard rate of the TD distribution is attractively flexible therefore it is useful and

suitable for many real life situations. Dagum distribution is the special case of the TD

distribution when . Domma (2002), Domma, Latorre and Zenga (2011) and

Domma, Giordano and Zenga (2011) using Glaser’s theorem (1980) proved the

proposition of the hazard rate function of the Dagum distribution. So taking these

propositions and Glaser’s theorem (1980), we concentrate on the additional parameter

and find out the following four behaviour of the hazard rate function on different

combinations of the parameters.

Haz

ard r

ate

Figure 4.10: The behaviour of the hazard rate of TD distribution for various

parameters values such as: 1.0[0.5]1.0, 1.0[0.5]1.0,

0.75[0.25]1.75 and 0.8[0.1]-0.4, 0.75[0.25]1.75 and


1. The hazard function of TD distribution is decreasing if

a) ⁄ and

b) ⁄ and

81

c) [ ⁄ ] and

2. It is upside down bathtub (increasing-decreasing) if

a) ⁄ and

b) ⁄ and

3. It is bathtub and upside down bathtub if

a) (

) and

b) ⁄ and

4. It is upside down bathtub if

5. (

) and

6. ⁄ and

Figure 4.10 shows the hazard function pattern of TD distribution with various choices

of parametric values

4.3.4 Order statistics of the transmuted Dagum distribution

Let be a random sample of size from the TD distribution and let

denote the corresponding order statistics. Then the pdf of the

ordered statistics follows the TD distribution is derived as follow

( ) [ ( )]

( )

0 ( (

)

)

1 ∑ (

)

( (

)

)

Let suppose that the smallest value also follows the TD distribution, and then the pdf

of the smallest order statistic, is obtained as

( ) [ ( )]

( )

0 ( (

)

)

1

∑ ∑ ( )

(

)

( (

)

)

Generally the pdf of order statistics is given by

( )

[ ( )]

[ ( )]

( )

82

0 ( (

)

)

1

∑ ∑ (

)

(

)

( (

)

)

Sometimes interest is in the joint pdf such as to find the joint breaking strength of

certain equipment, for the TD distribution the pdf of and , when 1 r s n

is derived as follows

(

)

0 ( (

)

)

1

∑ ∑ ∑ ∑ (

)

(

) (

) ( )

( (

)

)

( (

)

)

where

4.3.5 TL-moments

According to the definition of the generalized TL-moment, defined in Section 4.2.4,

the th generalized TL-moment with lowest and highest trimming is derived for

the TD distribution as given follows

∑ ∑ ∑ (

) (

)

(

)

(4.28)

⁄

0 [ ⁄ ]

[ ]

[ ⁄ ]

[ ]1

The L-, TL-, LL- and LH-moments are the special cases of the generalized TL-

moment. These moments are obtained by considering, for L-moments,

83

for TL-moments, for LL-moments and for LH-

moments in (4.28), and respectively given by

∑ ∑ ∑ (

) (

)

(

)

⁄

0 [ ⁄ ]

[ ]

[ ⁄ ]

[ ]1

∑ ∑ ∑ (

) (

)

(

)

⁄

0 [ ⁄ ]

[ ]

[ ⁄ ]

[ ]1

∑ ∑ ∑ (

) (

)

(

)

⁄

0 [ ⁄ ]

[ ]

[ ⁄ ]

[ ]1

and

∑ ∑ ∑ (

) (

)

(

)

⁄

0 [ ⁄ ]

[ ]

[ ⁄ ]

[ ]1

The preferable value of and is upto 4 for LL- and LH-moments.

4.3.6 Parameter estimation

To estimate the parameters of the TD distribution by MLE, let suppose that

be independently distributed random variables of size . Then the sample

likelihood function for this distribution is given as

(

)

∏ ⁄

[ ⁄ ]

84

The sample log-likelihood function corresponding to the above expression is obtained

as

∑

(4.29)

∑ ⁄

∑ [ ⁄ ]

To find the ML estimates, and , respectively, we differentiated (4.29) with

respect to the parameter and equate them equal to zero, we get

∑

∑ ⁄ ⁄

⁄

∑ ⁄ ⁄

[ ⁄ ] ⁄

(4.30)

∑

⁄

⁄

∑ ⁄

[ ⁄ ] ⁄ ⁄

(4.31)

∑ ⁄

∑ ⁄ ⁄

[ ⁄ ]

(4.32)

and

∑ ⁄

[ ⁄ ]

(4.33)

The exact closed forms solution to derive the estimator for unknown parameters is not

possible, so the estimates ( ) are obtained by solving the above four

nonlinear equations simultaneously. This solution of the nonlinear system is easier by

Newton-Raphson approach.

4.3.7 Application

In this section, we have compared the performance of the TD distribution with Dagum

distribution by considering the monthly maximum precipitation data of Islamabad, the

capital city of Pakistan. The geographical location of this city has Latitude 33.71

North and Longitude 73.07 South with humid subtropical climate and has five

85

seasons. This area receives heavy rainfall during monsoon season. The data of

monthly precipitation retrieved from the Regional Meteorological Center (RMC)

Lahore and Pakistan Meteorological Department (PMD) Islamabad. The length of

data is 640 recorded from January 1954 to December 2013 excluding some

unobserved or unreported months and the summary statistics are given in Table 4.4.

Table 4.4: Summary Statistics for monthly maximum precipitation data of the

Islamabad, Pakistan

Length Average Minimum Maximum Q1 Median Q3 S.D

640 80.90 0.10 641.00 20.35 49.90 101.90 94.98

Q = Quartile, S.D = Standard Deviation

In order to compare the TD with its parent distribution, we consider criteria of ,

AIC, AICC, BIC and KS goodness of fit test for the precipitation data set. The good

fitted distribution have the minimum value of , AIC, AICC , BIC and KS than the

others.

It is better to test the superiority of the TD distribution over the Dagum distribution

before analyzing the data. We employed the LR test statistic here as it is applied in

TSM distribution. So the computed value of LR test statistic is . We

observe that the , so we reject the null hypothesis and found that

the TD model is best for the data set.

Table 4.5: Estimated parameters of the TD and Dagum distribution for precipitation

data set

Model Parameter

Estimate AIC AICC BIC K-S

Transmuted

Dagum

3452.71 6913.42 6913.48 6931.27 0.0280

Dagum

3464.08 6934.16 6934.20 6947.544 0.1646

Variance-covariance matrix of the MLEs under the TD distribution is computed as

86

1

0.0407 2.7555 0.0097 0.0094

2.7555 1360.2 0.5489 13.361

0.0097 0.5489 0.0028 0.0004

0.0094 13.361 0.0004 0.4924

I

Thus, the variances of the ML estimates are, 0.2019, ( ) 36.8808,

0.0527 and ( ) 0.3863. Therefore, 100 1 0.05 % confidence

interval for , , and are [1.8240, 2.6156], [60.657, 205.23], [0.2946, 0.5015]

and [-0.4007, 1.1136], respectively.

The results of Table 4.5 indicates that the proposed TD distribution fits well as it has

the smallest ( ;.)x , AIC, AICC and BIC as compared to the Dagum distribution.

The KS goodness fit test is also employed to evaluate the best-fitted model for the

precipitation data. The calculated value of this test is 0.0280, whereas the tabled

critical two-tailed values at 0.05 and 0.01 significance levels are 0.0538 and 0.0644

respectively. According to Sheskin (2003), if the value of is greater or equal to the

critical value the null hypothesis is rejected. Thus the null hypothesis cannot be

rejected for the TD distribution as the value of the is not greater or equal to the

critical values.

Figure 4.11: Empirical, fitted TD and Dagum cdf of the precipitation data set and

maximum distance highlight.

87

Figure 4.12: PP-plots for fitted TD and Dagum distribution

Both empirical cdf and PP-plots also indicate that the TD distribution is better than its

competitor Dagum distribution to model the precipitation data set. As we noticed that

the TD distribution follows the empirical pattern of the data very closely and similarly

in PP-plot, the TD distribution lies almost perfectly on the 45o line. These all results

lead us to conclude that the TD distribution is a better model than the parent model for

fitting on such data.

Table 4.6: First four sample moments, , and of C-moments, L- and

LT-moments for precipitation data set

Moments L-moments TL-moments

1st 80.90218 80.9022 62.8439

2nd

15381.66 43.9082 20.3095

3rd

4.72×106 18.0583 5.60989

4th

1.89×109 10.0589 2.42731

1.16193 1.84253 3.09429

2.47134 0.41127 0.27622

The sample moments and their ratios are presented in Table 4.15. The average

monthly precipitation in Islamabad is 80.9022 with the full data set and 62.8439 for

trimmed data set. Similarly asymmetry and peakness are high for the full data set and

if the extreme values are trimmed then it reduced.

88

4.4 Transmuted New distribution

The New distribution has been proposed recently by Sarhan, Todj and Hamilton

(2014) as a simple and useful reliability model for analyzing the lifetime data. The

hazard rate of the New distribution is upside down bathtub. Here, we introduced the

generalization of the New distribution and called it as transmuted New distribution.

We refer the New and transmuted New distribution as and

respectively. The shows a variety of shapes which enable it to model

various data sets of different fields. It is also good to model the extreme value

frequency data as well.

Consider a random variable , follows the , with the following pdf and cdf

[ ] (4.34)

and

[ ] (4.35)

respectively.

A random variable have a , if the cdf and pdf are derived using (2.16) and

(2.17) respectively. The cdf of the is given as follows

(

)

[ ][

]

(4.36)

and the pdf obtained as

(

)

[ ][

]

(4.37)

The is more flexible and capable to model the complex lifetime data as

compared to parent distribution. The parent distribution is the special case of the

when . To illustrate the flexibility of the , various shapes of

the pdf and cdf are sketched in Figure 4.13 respectively, assuming some possible

parametric values.

89

Figure 4.13: The pdf’s of for various values of parameters:

1.0[0.5]3.0, 0.5; 2, 1.0[0.5]1.0 with solid, dashed, dotted, dotdash

and longdash lines respectively.

It is observed that the shape pattern of the mostly depends on the transmuted

parameter . The parent distribution is only J-shaped, now showed the

moderate to high positive skewed and J-shaped pattern. The various shape of the cdf

of are illustrated in Figure 4.13 assuming some possible parametric values.

Figure 4.14: The cdf’s of for various values of parameters:

1.0[0.5]3.0, 0.5; 2, 1.0[0.5]1.0 with solid, dashed, dotted,

dotdash and longdash lines respectively.

90

4.4.1 Reliability analysis of the transmuted New distribution

This distribution is considered as the lifetime distribution, so its reliability analysis

has worth importance, as the reliability function provides the probability of survival

beyond failure time. The reliability function of the is derived as

[ ][

]

(4.38)

The shape of the reliability function is presented in Figure 4.15.

Figure 4.15: Shapes of Reliability function of with various choices

of parametric values.

Hazard rate function is also an important measure in reliability analysis; it quantify

the decline towards failure rate. The hazard rate function of is derived using

the usual formula and we get

[ ][ ]

[ ][ ] (4.39)

4.4.2 Moments

The ordinary moment of the is derived as

∫

91

(

)

∫

[ ][

]

0 { }

1

(4.40)

.

/

where .

The mean of the is obtained by setting in (4.40) as

[ ]

The , and can be obtained using first four ordinary moments.

The random numbers for can be generated by taking the following steps

1. Generate the uniform distribution within the variation 0 and 1.

2. Determine the required sample size

3. Specify the value of the population parameters

4. Replace the parameters value and uniform numbers ( )u in (4.41)

(

)

[ ][

]

(4.41)

5. Finally, numerically solve the (4.41) for to find the random numbers those

follow the transmuted New distribution.

4.4.3 Order Statistics

Here we assume that is a random sample from with

pdf and cdf given in (4.37) and (4.36) respectively. Let be

the order values of the preceding sample in non-decreasing order of magnitude.

The order statistics of , is given by

[ ]

(

)

∑ ∑ (

) (

)

[ ]

92

The first order statistics of , is given by

[ ]

(

)

∑ (

)

[ ]

[ ]

Generally the pdf of the order statistics for is as follows

[ ] [ ]

∑ ∑ ∑ (

) (

) (

)

[ ]

4.4.4 Estimation

In this section, the MLE procedure for the transmuted distributions is defined to

estimate the parametric values.

Let be an independently identically distributed (iid) random sample of

size from . Then, on the base of observed sample size the likelihood

function is defined as

∏[ ]

∏[

] ( ∑

)

(4.42)

The sample log-likelihood function corresponding to (4.42), is given by

∑

∑ [ ]

∑ [ ]

(4.43)

The following equations are the first order derivations of the (4.43) with respect to

parameters and the simultaneous solution of these set of equations yields the

parameter estimates

93

∑

∑ [

]

∑ 0 ( )

1

(4.44)

∑ 0

1

(4.45)

The exact solution for unknown parameters is not possible analytically from the

normalized equations. Therefore, the parameter estimates are obtained by solving

nonlinear equations simultaneously.

4.4.5 Application

In this section, the analysis of two data sets is illustrated to prove the superiority and

applicability of the proposed transmuted model. The first data set consists of 76

observations of the life of fatigue fracture of Kevlar 373/epoxy that are subject to

constant pressure at the 90% stress level until all had failed. This data set is easily

available in Andrews and Herzberg (1985) and Gómez, Bolfarine and Gómez (2014)

studies. The second data set is about the maintenance actions for the number 4 diesel

engine of the U.S.S. Grampus, it consists of 57 observations, for details see Meeker

and Escobar (1998).

The proposed transmuted and parent distributions are fitted on both data sets. The

parameters of each model are estimated by MLE and the estimated results (estimates,

standard error and confidence interval) are provided in Table 4.7.

Table 4.7: Parameter estimates, S.E and C.I of two data sets

Model Parameter

Estimate S.E C.I

Lower Upper

Life of fatigue fracture

0.60030 0.06600 0.47093 0.72966

0.7976 0.15978 1.0000 0.4844

0.42129 0.04769 0.32780 0.51478

Maintenance Actions

0.13603 0.01511 0.10642 0.16564

1.000 0.19691 1.0000 -0.6762

0.09362 0.01103 0.07198 0.11526

94

To check the statistical superiority of the over the , the LR test is

employed to test the hypothesis versus . The value of the LR

statistic for the data set life of fatigue fracture and maintenance actions are 5.811 and

12.269 respectively. So we reject the null hypothesis as each value is greater than the

3.841. Furthermore, we considered the , AIC, AICC and BIC

goodness of fit criteria for the sake of comparison to verify the better fitted model.

These goodness of fit measures are given in Table 4.8.

Table 4.8: Goodness of fit measure for transmuted and parent distributions

Model AIC AICC BIC K-S

Life of fatigue fracture

244.063 248.227 248.227 252.725 0.1000

249.874 251.874 251.928 254.205 0.1271

Maintenance Actions

337.0020 341.0020 341.2242 345.0881 0.1122

349.2712 351.2712 351.3439 353.3143 0.1463

Figure 4.16: The empirical and fitted cdfs of the life of fatigue fracture data

95

Figure 4.17: PP-plots of the fitted distribution of the life of fatigue fracture data

Figure 4.18: The empirical and fitted cdfs of the life of maintenance data

Figure 4.19: PP-plots of the fitted distribution of the maintenance data

96

For both the data sets, we plotted the empirical and fitted cdf plots and PP-plot. Figure

4.16, Figure 4.17, Figure 4.18 and Figure 4.19 show these plots and indicate that the

is fit and model the data better than .

4.5 Conclusion

Three transmuted distributions named as transmuted Singh-Maddala, transmuted

Dagum and transmuted New distribution are proposed in this study as the

generalization of the parent distribution. These distributions are quite flexible, and

their application diversities increased due to the additional transmuted parameter as

compared to the standard distribution. To show the flexibility of new densities the

plots of the pdf, cdf, reliability function and hazard functions are sketched. The

moments and other essential properties of the proposed distributions are derived. The

densities of the lowest, highest, th order statistics, the joint density of the two order

statistics and TL-moments are also studied. The parameter estimation is obtained by

the maximum likelihood estimation via Newton-Raphson approach. To evaluate its

worth five goodness of fit criterion are considered for the selection of most

appropriate model. Based on criteria, the results of real life data sets have showed that

transmuted distributions are superior to the base distribution. Finally, we hope that the

proposed model will serve better in income distribution, actuarial, meteorological and

survival extreme value data analysis.

97

5. CHAPTER 5

Extreme Value Analysis by Double Bounded Transmuted

Distributions

5.1 Introduction

The continuous double-bounded variables are fairly popular in economics,

meteorological, hydrological, civil engineering, social and behaviour studies. It is

quite difficult and unrealistic to analyze such type of data using Gaussian theory

models. An alternative approach is to use the double-bounded probability

distributions to obtain the accurate results. An interesting aspect of the bounded

distribution is that, these distributions have the hazard rate function either increasing

or bathtub shaped. Several reliability or survival analysis studies require such kind of

hazard rate function.

In hydrology, the double bounded distributions play a significant role in modeling the

flood data, because the characteristics of the flood, such as volume and duration, may

be as important as peak flow. These characteristics may also be analyzed by

probabilistic logic in many situations, (for detail see Fernandes, Naghettini and

Loschi, 2010). Mukherjee and Islam (1983) and Moore and Lai (1994) described that

the designed lifetime experiments have only specific range rather than the infinite

range. Therefore, the selected failure-time distribution must be capable of modeling

the failure rate over any limited interval, especially in survival analysis.

It is observed that economic, manufacturing and servicing tasks generate bounded

data. As these tasks are performed periodically and must be completed before an end-

to-end deadline. Each task is bounded to execute within financial constraints and on a

particular processing element. Therefore, to model these types of data by bounded

distribution will provide the precise results. Middleton (1997) estimated the

distribution of demand using bounded sales data through probability distribution,

98

considering the flight capacity data, as the airline planning department can sell the

passenger tickets within a fixed number of seats available. To model the data that

appears from such situations, transmuted bounded distributions are proposed in this

study. These proposed densities will accommodate the double-bounded data,

especially the datasets those are defined in the continuum between 0 and 1, in a better

way.

Rest of the Chapter proceeded as follows. In Section 5.2, the transmuted

Kumaraswamy distribution is developed using QRTM and the pdf and cdf are derived

for this distribution. In Subsection 5.2.1, basic statistical properties of the transmuted

Kumaraswamy distribution are studied. The Subsection 5.2.2 is about the reliability

and hazard rate function of the distribution. The order statistics of this distribution

defined and the densities of lowest, highest, and joint ordered statistics have been

derived in Subsection 5.2.3. In Subsection 5.2.4, we have derived the generalized TL-

moments and its special cases. To estimate the unknown parameters of the new

proposed density, the method of MLE is discussed and in addition, the information

matrix is determined in Subsection 5.2.5. In Subsection 5.2.6 the empirical study is

carried out using three real life examples and found out transmuted Kumaraswamy

distribution is more advantageous than the parent distribution.

Section 5.3 presents transmuted Power function distribution along with its graphical

shapes of the density and distribution function. In Subsection 5.3.1, mathematical

properties are given such as C-moments, moment generating function, and mode of

the distribution. The Subsection 5.3.2 is about the quantile function and random

number generation. Reliability and hazard functions are derived and given in

Subsection 5.3.3, with their graphical presentation. The order statistics of this

distribution have been explored in Subsection 5.3.4. The generalized TL-moments

with its special cases are obtained, and maximum likelihood parameter estimation

approach is employed in Subsection 5.3.5 and 5.3.6 respectively. A simulation study

is carried out in Subsection 5.3.7. In Subsection 5.3.8, two real data sets have been

considered to exemplify the application and comparison of the transmuted Power

function distribution with parent distribution. Finally, we concluded our study in

Section 5.4.

99

5.2 Double bounded transmuted Kumaraswamy distribution

The Beta distribution has been used for many years for the datasets; those are

restricted in finite interval. A two parameter double-bounded Kumaraswamy density

was proposed by Kumaraswamy (1980) and had been widely used as an alternative to

the Beta distribution with tractability advantages. It was initially proposed for

hydrology data analysis. This distribution belongs to the first kind of McDonald’s

(1984) generalized Beta distribution. According to Silva and Barreto-Souza (2014)

and Mitnik and Baek (2013), the Kumaraswamy (Kw) distribution is more appealing

due to its close-form cumulative distribution function and more efficient and easier in

implementation than the Beta distribution in various disciplines. Sundar and Subbiah

(1989) described the ocean waves height by the Kw distribution and found it better

than the Rayleigh and Extreme value distribution. Fletcher and Ponnambalam (1996)

also modelled the reservoir capacity and storage state variable by the Kw distribution.

Zhao et al. (2013) fit the adjusted Kw distribution on the consumer attitudes toward

genetically modified products and consumer welfare. According to Nadarajah (2008)

and Koutsoyiannis and Xanthopoulos (1989), many studies in the hydrological

literature have considered this distribution because it considered as a better alternative

to the Beta distribution due its simple closed form of the cdf. The pdf of the two

parameters Kw random variable is given by

(5.1)

and its cdf is

. (5.2)

To further enhance its scope to model double-bounded data, we have proposed

transmuted Kumaraswamy distribution in this study. Indeed, it is of great interest to

model the finite interval data of various disciplines by double-bounded model. The

transmuted Kumaraswamy (TKw) distribution is obtained using the methodology of

Shaw and Buckley (2009). The cdf of the TKw distribution is given by

[ ][ ]

| | (5.3)

which on differentiation yields the following pdf of TKw distribution

[ ] (5.4)

100

The parent (Kw) distribution is a special case of TKw distribution when . TKw

density function shows a variety of behavior those are graphed in Figure 5.1,

assuming some possible parametric values.

It is observed analytically and graphically TKw distribution shows various shapes for

different combination of the parameters. These are moderate to highly skewed,

symmetric, unimodal and uniantimodal. So it is capable to model various types of

double-bounded data.

Figure 5.1: The pdf’s of TKw distribution for various choice of parameters:

0.5, 0.5,1[1]4.0, 0.1; : 2.5[1]6.5, 2.0[0.25]3.0, 0.8; :

2.0, 2.0[1]7.0, 0.5; : 3.0, 4.0, 1.0[0.5]1.0; with

solid, dashed, dotted, dotdash and longdash lines respectively.

It is also noted that the TKw density has the following shape properties,

a) 1 and -1 +1 U-shaped uniantimodal;

b) 1 and -1 0 increasing;

c) 1 and 0 1 decreasing;

101

d) 1 and 0.5 uniform;

e) 1, 1 and -1 +1 exponentially increasing;

f) 1, 1 and -1 +1 exponentially decreasing;

g) 1 and 0 unimodal positively skewed;

h) 1 and 0 unimodal symmetric to negative skewed and

i) 1 and 0 constant.

The cdf of the TKw distribution for various combinations of the parameters sketched

in Figure 5.2.

Figure 5.2: The pdf’s of TKw distribution for various choice of parameters:

0.5, 0.5,1[1]4.0, 0.1; : 2.5[1]6.5, 2.0[0.25]3.0, 0.8; :

2.0, 2.0[1]7.0, 0.5; : 3.0, 4.0, 1.0[0.5]1.0; with

solid, dashed, dotted, dotdash and longdash lines respectively.

102

5.2.1 Basic statistical properties

This section is about the fundamental statistical properties including proper density

function, moments, moment generating function, Quantiles and random number

generation for the TKw distribution.

5.2.1.1 Proper density function

Lemma: The given in (5.4) is a proper probability density function.

Proof: is the non-negative function and the integration of this function

over the full support (0,1) is equal one.

∫

∫ [ ]

∫

∫

Using simple transformation and beta function ∫

, we

obtain the lemma that the TKw density is a proper probability density function.

5.2.1.2 Moments

The moment of the TKw distribution is defined as follow

∫

∫ [ ]

Using beta function, we get

(

) * ,(

) (

)⁄ -+ (5.5)

Figure 5.3: The mean plot of the TKw distributions with respect to the parameters

103

First four moments about origin can be obtained by substituting 1,2,3 and 4 in

(5.5). The variance, , and can be obtained using the usual formulas.

5.2.1.3 Moment generating function

The moment generating function of random variable of TKw distribution is given

by

∫

∫ .

/

∑

(

) * ,(

) (

)⁄ -+

5.2.1.4 Quantile function

The quantile function of the TKw distribution is obtained by inverting the

distribution function in the following form

[ ( √

)

⁄

]

⁄

(5.6)

In particular, the median of the TKw distribution is obtained from (5.6) and given

below

* {( √ ) ⁄ } ⁄

+ ⁄

.

5.2.1.5 Random number generation

The quantile function is often used to generate the random sample, which follows the

particular a probability distribution for the simulation study. If follows a standard

uniform distribution, then using the inversion method the random variate

[ ( √

)

⁄

]

⁄

is come from the TKw distribution that can yield random numbers when parameter

are considered known.

104

5.2.2 Reliability and hazard rate function

The reliability function quantifies the probability that a component or a system will

continue its work without failure during prescribed time interval . It can state as,

the reliability function is the probability of survival beyond failure time such

as . The cdf

used to find the reliability function, such as

. The reliability function of a TKw distribution is given by

[ { }] (5.7)

Hazard rate measures the decline towards failure rate. The probability approaches to

failure increases as the value of the hazard function increase. Hazard rate is the ratio

of density function and the reliability function . The hazard function of

TKw distribution is defined as

[ { }]

[ { }] (5.8)

5.2.3 Order statistics of transmuted Kumaraswamy distribution

We derived the pdf of the , and order statistics for TKw distribution in

this section . Let is a random sample from TKw distribution with pdf

and cdf given in (5.4) and (5.3) respectively. Let be the order values

of the preceding sample in non-decreasing order of magnitude.

Let the distribution of the minimum statistics follow the TKw distribution, and then

the smallest order statistic, has the pdf in the following

form

( ) ∑ ∑ ∑ (

) ( ) (

)

( )

* , (

)

-+

The density for the order statistics of TKw distribution,

is given by

( ) ∑ ∑ (

) (

)

( )

* , (

)

-+

105

Generally the density distribution of the order statistics with the TKw distribution

is obtained as

( )

∑ ∑ ∑ (

) (

) (

)

(

)

* , ( )

-+

Let the two order statistics and have the common TKw density and are

independent assuming . Now consider and then the

joint density of order statistics is given as

∑ ∑ ∑ ∑ ∑ ∑ (

) (

) (

)

(

) (

) (

)

where

[ { }]

and

[ { }].

In this section, size of the sample should be small or moderate because due to very

large sample size convergence may be an issue.

5.2.4 Generalized TL-moment and its special cases

In this section, the generalized TL-moment and its special cases for the TKw

distribution are derived. The generalized TL-moment for TKw distribution is given

below

∑ ∑ ∑ ∑ (

) (

)

(5.9) (

) (

)

. ⁄ ( )

⁄ / 0 .

⁄

⁄ /1

106

As already mentioned that the L-, TL-, LL-, LH-moments are the special cases of the

generalized TL-moment. These special cases for TKw distribution are obtained from

(5.9) and are provided in the following form, respectively.

∑ ∑ ∑ ∑ (

) (

)

( ) (

)

(5.10)

.

⁄ ( )

⁄ /

0 . ⁄

⁄ /1

∑ ∑ ∑ ∑ (

) (

)

(

) (

)

(5.11)

.

⁄ ( )

⁄ /

0 . ⁄

⁄ /1

∑ ∑ ∑ ∑ (

) (

)

(

)

(5.12) (

)

.

⁄ ( )

⁄ /

0 . ⁄

⁄ /1

∑ ∑ ∑ ∑ (

) (

)

(5.13) (

) (

)

. ⁄ ( )

⁄ / 0 .

⁄

⁄ /1

107

5.2.5 Estimation and Information Matrix

In this section, the interest is to estimate the parameters of TKw distribution by the

method of MLE. Let be i.i.d random variables of size . Then the

likelihood function for this distribution is found as

∏

[ ] (5.14)

and the sample log-likelihood function as

∑[

( )]

(5.15)

The partial derivatives with respect to parameters , and are obtained as follows

∑ 0

1

(5.16)

∑ 0

1

(5.17)

and

∑ 0

1

(5.18)

The exact solution to derive the estimator for unknown parameters is not possible

analytically, so the estimates ( ) can be obtained by solving the above three

nonlinear equations simultaneously. The observed information matrix is given by

[

]

and the elements of this matrix are derived to obtain numerical results.

To find the estimates of parameters by maximizing the likelihood equations is often

impossible due to the nonlinearity. Therefore, Newton-Raphson, an iterative

procedure is considered, that is a powerful technique for solving equations

numerically.

108

5.2.6 Empirical Study

Empirical studies are presented in this section, to compare the TKw and Kw

distributions. In three considered real data sets, the first data set is about the Muslim

population percentage in different countries. The second data set is the annual

maximum peak flows at Kalabagh site on Indus River, Pakistan and the third data set

is about the daily ozone level measurement in New York City. The detail description

of these data sets is given below. The method of MLE is used to estimate the

unknown parameters. Finally, the models are compared by the goodness of fit

criteria, and visual comparisons are also provided.

Dataset 1: Total countries in the world are 227 in which 75 are wholly non-Muslim,

and 3 are entirely populated with the Muslim population. So in the first data set, we

consider the percentage of Muslim population in the remaining 149 countries. This

dataset retrieved from http://www.qran.org/a/a-world.htm and was based on 2004

Census projection. This data set was previously used for Kw distribution by Silva and

Barreto-Souza (2014).

Dataset 2: The second example of the study deal with the annual maximum peak

flows at Kalabagh site located on the Indus River, Pakistan. It is the longest river in

Pakistan, and the Kalabagh site is on latitude 32.95 North and longitude 71.50 South.

The maximum design capacity of this site is 950000. The annual maximum peak

flows have been retrieved from the hydrology department, Water and Power

Development Authority (WAPDA) Lahore and Pakistan Meteorological Department

(PMD) Islamabad. The data has 85 observations, recorded from 1928 to 2012 and

presented in Figure 5.4. To transform the data the following equation is used and it is

postulated by Kumaraswamy (1980) to obtain the double bounded data.

⁄ (5.19)

Where and are the lower and upper water capacity bound of the data

respectively.

http://www.qran.org/a/a-world.htm

109

Figure 5.4: Time series plot of Annual maximum peak flow at Kalabagh site.

To investigate the randomness of the data set a time series plot, and non-parametric

Bartels run test has been applied. The visual inspection and corresponding p-value

(0.1093) of the run test are acclaimed that the hypothesis of randomness at 5% level

of significance cannot be rejected. So it seems logical to study this data through

probabilistic models.

Dataset 3: The third data set is the daily ozone level measurement in New York City,

the data and its description available in the Nadarajah (2008) and in his study he

considered equation (5.19) for transformation purposes.

In order to compare the TKw and Kw distributions, log-likelihood (ℓ), AIC and

goodness of fit test are considered for all three data sets.

These results of data set 1, 2 and 3 are given in Table 5.1. The better model normally

has the comparatively minimum value of these criteria. We can also employ LR-test

to test whether the fitted TKw distribution for given data sets is statistically “superior”

to the fitted Kw distribution. To perform this test the maximized restricted and

unrestricted log-likelihoods can be computed under the null and alternative hypothesis

(restricted, Kw model is true) and (unrestricted, TKw model is

true). The value of the test statistics are computed for three data sets and obtained,

6.96, 9.49 and 4.23 respectively. These all are greater than corresponding table

values ( ). Thus, the results do not support the null hypothesis and found

that the TKw model is the best for all three datasets.

110

Table 5.1: The estimates, standard error of estimates, confidence interval and

goodness fit criteria for three real data sets

Model Parameter

Estimate S.E C.I

AIC Lower Upper

Data Set 1 (Percentage of Muslim population)

TKw

0.0355

0.0809

0.1844

0.2390

0.3944

0.0448

0.3782

0.7115

0.7676

-234.47 -228.47 0.1417

Kw

0.0376

0.0833

0.2174

0.4126

0.3648

0.7392 -227.52 -223.51 0.2119

Data Set 2 (Kalabagh)

90

TKw

0.3326

0.7143

0.1448

2.8659

1.8250

0.4929

4.1698

4.6251

1.000*

-75.908 -69.608 0.0525

Kw

0.3046

0.7062

2.5097

2.6722

3.7037

5.4407 -66.412 -62.412 0.0600

Data Set 3 (Ozone level)

TKw

0.1092

0.8255

0.2824

1.0204

1.2165

0.1534

1.4484

4.4524

1.000*

-117.18 -111.18 0.0923

Kw

1.002

0.5808

0.9375

2.6637

1.3301

4.9405 -112.90 -108.79 0.0958

*Approximately approaches to maximum value of

The results in

Table 5.1 indicates that the proposed TKw distribution fits well as it has the smallest

, AIC and statistic for all data sets in comparison with Kw distribution.

111

Figure 5.5: PP-plot, empirical, fitted TKw and Kw cdf for data set 1


112


Both Ecdf of distance and PP-plots also indicate that the TKw distribution is better

than its competitor Kw distribution for all data sets. So according to all these

goodness of fit criteria, the new double-bounded TKw distribution provides a better

fit than the Kw distribution. Therefore, we hope that the new double-bounded

distribution will provide a flexible environment and better fitting of double-bound

data in various disciplines.

L-moments and its related moments are used to compute the fundamental

characteristics of a dataset in a better way than C-moments and to show the true

picture of the data. The first and second moment of these moments describe the

average and variation in the data, respectively. Consistency, symmetry and

peakedness are calculated using the 2nd

, 3rd

and 4th

moments. These moments and

moment ratios are calculated using (5.10), (5.11), (5.12) and (5.13) and reported in

Table 5.2 for all considered datasets.

113

Table 5.2: First four moments, and of C-, L-, TL-, LL- and LH-moments

by three data sets

C-moments L-moments TL-moments LL-moments LH-moments

Data Set 1 (Percentage of Muslim population)

1st 0.4951 0.4951 0.4841 0.5230 0.4673

2nd

0.3660 0.0279 0.0116 0.0292 0.0126

3rd

0.2772 0.0110 0.0050 0.0131 0.0016

4th

0.2143 0.0086 0.0020 0.0057 0.0051

0.7022 0.0563 0.0240 0.0558 0.0270

-0.5639 0.3942 0.4310 0.4486 0.1269

Data Set 2 (Kalabagh)

1st 0.8170 0.8170 0.7480 1.0763 0.5575

2nd

0.5123 0.2594 0.1312 0.2463 0.1428

3rd

0.0340 0.0690 0.0259 0.0732 0.0188

4th

0.2355 0.0408 0.0130 0.0345 0.0164

0.4822 0.3175 0.1754 0.2288 0.2561

-0.2142 0.2660 0.1974 0.2972 0.1316

Data Set 3 (Ozone level)

1st 0.3706 0.3706 0.3074 0.5555 0.1857

2nd

0.1499 0.1849 0.0938 0.1861 0.0913

3rd

0.7401 0.0632 0.0241 0.0611 0.0231

4th

0.0415 0.0285 0.0095 0.0257 0.0099

0.3034 0.4989 0.3051 0.3350 0.4916

6.4251 0.3418 0.2569 0.3283 0.2530

5.3 Double-bounded transmuted Power function distribution

Power function distribution is a quite familiar double-bounded distribution. It is

simple in use and commonly preferred in reliability analysis rather than the

mathematically complicated distributions. Mukherjee and Islam (1983) studied this

distribution as a finite‐range distribution for failure time data and considered it as a

reliable finite range (double-bounded) failure rate distribution. Lai and Mukherjee

(1986) discussed finite range Power Function (PF) distribution in more details, as a

finite range distribution and reported its reliability and failure rate properties

comprehensively.

Siddiqui and Mishra (1995) estimated the reliability and hazard rate function of this

double-bounded PF distribution through Bayesian framework. Dorp and Kotz (2002)

revisited standard two-sided PF distribution and introduced its application in financial

engineering. Herein, our motivation is to introduce a better version of PF distribution

with its application in extreme events data. Therefore, we have proposed the

114

transmuted Power function distribution, to provide better-fitted distribution for the

bounded data.

A positive random variable is PF-distributed if its pdf is given by:

(

)

(5.20)

and has the cdf in following form

(

)

(5.21)

where and are shape and scale parameters respectively.

Now we derive the transmuted distribution, to compare it with its parent (PF)

distribution. The cdf of the Transmuted Power Function (TPF) distribution is obtained

assuming QRTM and is given by

(

)

0 (

)

1 (5.22)

and its respective pdf is given as

(

)

0 (

)

1 (5.23)

where and are scale and shape parameters, respectively and is the transmuted

parameter with range . Without loss of generality, we may take 1 in

(5.22) and (5.23), now we derive the mathematical properties of the TPF distribution

by considering 1.

The PF distribution is a special case of the TPF distribution when . The TPF

distribution gained the variety of behavior in shape that PF distribution does not

attain. So the proposed distribution is more flexible to analyze diversity in real life

situation data. The pdf and cdf of TPF distribution are sketched for various

combinations of the parameters in the following Figure 5.8 and Figure 5.9

respectively.

115

Figure 5.8: The pdf of the TPF distribution for various values of the parameters:

0.3, 0.5, 3.0, 5.0 and 0.0; 0.5 and 0.1[0.2]0.7; 3.0,

0.3[0.2]0.9; 0.5, 3.0, 0.5, 3.0 and 0.7[0.4]0.7 with solid, dashes, dotted

and longdash lines respectively.

116

Figure 5.9: The cdf of the TPF distribution for various values of the parameters:

0.3, 0.5, 3.0, 5.0 and 0.0; 0.5 and 0.1[0.2]0.7; 3.0,

0.3[0.2]0.9; 0.5, 3.0, 0.5, 3.0 and 0.7[0.4]0.7 with solid, dashes,

dotted and longdash lines respectively.

5.3.1 Mathematical properties

In this section, statistical properties of TPF distribution are delivered.

Theorem 3.1: Let said to have a TPF distribution. Then the C-moment of the

is (

)

Proof. The C-moment is given by

∫ ∫ ( )

∫

∫

∫

This on simplification yield the th C-moment in the following form

117

(

) (5.24)

Then first C-moment, mean is obtained by assuming 1 in (5.24)

[

]

The variance is derived as given below

0

1

The and can easily obtain for TPF distribution through C-moments using

usual formulas.

Theorem 3.2: The moment generating function of , , when has a TPF

distribution is

∑

(

)

(5.25)

Proof: Let

∫

∫ .

/

Now, by taking simple steps we obtain the required result, given in (5.25).

Theorem 3.3: The mode of , when follows TPF distribution is

0

1

⁄

(5.26)

Proof : The mode, if it exists, is that value of for that 0 and 0. So

taking the first and second derivative of the (5.23) with respect to , we get

( )

118

The second derivative of the (5.23) is truly less than zero for all possible values of

and , now after equating first derivative equal to zero, we obtained the mode that is

given in (5.26).

5.3.2 Quantile function and random number generation

The quantile function of the TPF distribution is obtained by inverting the

distribution function given in (5.22), as given below

[

( √ )]

⁄

In particular, the median of the TPF distribution is obtained in the following form,

*( √ ) ⁄ + ⁄

The quantile function is often used to generate the random sample for a probability

distribution for the simulation study. If follows standard uniform distribution, then

following random variate

[

( √ )]

⁄

(5.27)

yield random data for TPF distribution when parameters are considered known.

5.3.3 Reliability analysis

The reliability function gives the probability of an item functioning for a specific

quantity of time without failure. The reliability function and cdf, are reverse of

each other. As and represent the probability of survival and probability of

failure, respectively. The reliability function of a TPF distribution is given by

( ) (5.28)

119

Figure 5.10: The reliability functions of various values choices of parameters:

0.3, 0.7; 0.3, 0.7; 5.0, 0.7 and 5.0, 0.7 with solid,

dashes, dotted and longdash lines respectively.

Hazard rate function is the ratio of pdf and the reliability function. It is also another

important property of a random variable. The hazard rate function for the TPF

distribution is obtained as given by

( )

(5.29)

Figure 5.11: The hazard functions of various values choices of parameters.

120

It is noticed, the TPF distribution provides more flexible hazard rate function than PF

distribution. So, the attractive shapes of the hazard rate function are more useful in

many disciplines for survival or reliability analysis, as Figure 5.11 shows increasing

and bathtub-shaped hazard rate. It is also realized that the TPF distribution is

relatively more appropriate for civil engineers and hydrologists, because of its

flexibility, simplicity, and applicability to measure the reliability of the systems and

dams infrastructure.

5.3.4 Order statistics

In this Section, we derived the density functions for , maximum and minimum

order statistics for TPF distribution. Suppose be the continuous

ascending order sample then the pdf of the order statistics for TPF distribution is

obtained as follows

∑ ∑ (

) (

)

(5.30)

( )

Therefore, the pdf of the largest order statistics is given

by

∑ (

)

( )

and the pdf of the smallest order statistic is obtained as

follows

∑ ∑ (

) (

)

( )

5.3.5 Generalized TL-moment

In this section, generalized TL-moments for the TPF distribution are derived. Using

the generalized TL-moment, its special cases such as L-, TL-, LL- and LH-moments

are also derived and defined.

121

Let be a continuous random variable of sample size and its corresponding ordered

sample is denoted by . The generalized TL-moment for

TPF distribution is given as follows

∑ ∑ ∑ (

) (

) (

)

[

]

(5.31)

The special cases of the generalized TL-moment, L-, TL-, LL- and LH-moments are

listed as follows

∑ ∑ ∑ (

) (

) (

)

(5.32)

[

]

∑ ∑ ∑ (

) (

) (

)

(5.33)

[

]

∑ ∑ ∑ (

) (

) (

)

[

]

(5.34)

and

122

∑ ∑ ∑ (

) ( ) (

)

[

]

(5.35)

respectively.

5.3.6 Parameter Estimation

Let be an independent identical distributed random variable of size from the

density of TPF distribution given in (5.23). Then, the observed sample likelihood

function is defined as

∏

∏ 0 (

)

1

(5.36)

The observed sample log-likelihood function is given by

∑

∑ 0 (

)

1

(5.37)

Now taking the derivative of log-likelihood function with respect to the parameters

considering is the maximum value of the data and equate them equal to zero,

we get

∑

∑

[ ⁄ ]

∑ ⁄

[ ⁄ ]

The exact close form of maximum likelihood estimators is not possible, so the

estimates ( ) of parameters are obtained by solving the above nonlinear

equations. The solution of nonlinear system of equations is conveniently possible by

quasi-Newton algorithm. This algorithm numerically maximizes the log-likelihood to

estimate the parametric values. Considering the large sample approximation

properties the 100 two sided confidence interval for and are as

following

⁄ √ ( ) and ⁄ √

( ) ,

123

where ⁄ is the standard normal upper percentile and ( ) is the diagonal

variances from the variance-covariance matrix.

5.3.7 Monte Carlo Simulation study

In this section, a simulation study has been carried out to compare the performance of

the estimators with their corresponding true parameters. We generated 10,000 random

samples of size 10, 50, 100, 300) from the TPF distribution for different choices

of parameter 0.1, 0.4, 0.7, 0.9), 0.25, 0.50, 0.75) and using (5.27).

We have calculated the parameter estimates and the Mean Absolute Error (MAE) by

method of MLE. The results of this study are reported in Table 5.3.

Table 5.3: Estimates and MAE for different choices of parameter for TPF distribution

10n 50n 100n 300n estimate MAE estimate MAE estimate MAE estimate MAE

0.10

0.1284 0.0284 0.1192 0.0192 0.1001 0.1224 0.1000 0.0215

0.25

0.3160 0.0126 0.2743 0.0402 0.2591 0.0209 0.2548 0.0018

0.10

0.1135 0.0014 0.0965 0.0352 0.1004 0.0043 0.1002 0.0006

0.50

0.6017 0.0202 0.5317 0.0117 0.4937 0.0937 0.4993 0.0072

0.10

0.1137 0.0014 0.0979 0.0208 0.1032 0.0146 0.1011 0.0048

0.75

0.8409 0.0233 0.7909 0.0291 0.7616 0.2616 0.7584 0.0845

0.40

0.3619 0.0381 0.3774 0.0382 0.3984 0.0476 0.3994 0.0085

0.25

0.1613 0.1512 0.1907 0.2031 0.2323 0.2053 0.2458 0.1453

0.40

0.3638 0.0362 0.3827 0.0321 0.4043 0.0394 0.4024 0.0058

0.50

0.2309 0.3691 0.3408 0.2627 0.5097 0.1803 0.5042 0.1271

0.40

0.4179 0.0203 0.3832 0.0362 0.4047 0.0337 0.4019 0.0019

0.75

0.6402 0.0584 0.6961 0.3025 0.7537 0.1433 0.7521 0.0954

0.70

0.6618 0.0391 0.7043 0.0540 0.6994 0.0890 0.7001 0.0054

0.25

0.3273 0.1346 0.2992 0.2786 0.2391 0.2263 0.2451 0.1842

0.70

0.6620 0.0387 0.7143 0.0496 0.7011 0.0710 0.6994 0.0035

0.50

0.4007 0.1031 0.5632 0.1734 0.4928 0.1839 0.5035 0.1243

0.70

0.6445 0.0555 0.7030 0.0414 0.7044 0.0552 0.7023 0.0237

0.75

0.6811 0.1689 0.7491 0.1059 0.7513 0.1425 0.7506 0.0159

0.90

0.7420 0.2580 0.8094 0.1360 0.8846 0.0914 0.8945 0.0587

0.25

0.3829 0.2332 0.3348 0.2041 0.2618 0.1758 0.2549 0.1351

0.90

0.8149 0.0851 0.8697 0.0662 0.9103 0.0888 0.9013 0.0069

0.50

0.4084 0.0932 0.4350 0.1483 0.5105 0.1818 0.5094 0.1424

0.90

0.8324 0.0837 0.8721 0.0054 0.9054 0.0750 0.9007 0.1030

0.75

0.5835 0.3145 0.6443 0.0217 0.7499 0.1465 0.7501 0.0019

124

The MAE is calculated by ∑ | | ⁄ , where is the no of samples, is the

true parameter and is estimate of the parameter.

In Table 5.4, some more possible values of the parameters are considered such as

2, 10, 50), -0.50, 0.50).

Table 5.4: Estimates and MAE for different choices of parameter for TPF distribution

10n 50n 100n 300n Est.

* MAE Est. MAE Est. MAE Est. MAE

2.00 1.6323

236

0.5803

77

1.8874

9

0.6843

79

1.9125

3

0.4658

52

1.9260

4

0.5513

51 0.50

-0.641 0.5582 -0.591 0.3703

76

-0.435 0.4769 -0.470 -0.364

2.00 2.2299

7

0.4778

71

2.3495 0.4620

74

2.2992

6

0.6134

01

2.0095

8

0.6134

01 0.50 0.6713

89

0.3252

93

0.6088

72

0.4727

08

0.5850

67

0.4481

34

0.5234

55

0.3222

55 10.00

11.778

9

1.4699

1

8.7591

6

1.2676

5

10.737

9

1.6559

4

10.632

5

0.4856

72 0.50

-0.424 0.5045 -0.463 0.6734

2

-0.537 0.4788

08

-0.492 0.4088

92 10.00

9.2575

2

1.6576

5

9.5233

1

0.9277

24

9.2291

3

0.9156

28

10.379 0.5720

39 0.50 0.5175

19

0.6378

59

0.4311

41

0.6204

24

0.4829

74

0.4616

46

0.5134

24

0.3679

93 50.00

46.811 4.9180

1

50.181

3

3.0995 52.568

1

2.5827 50.787

5

2.9930

7 0.50

-0.389 0.6951 -0.652 0.4915

11

-0.530 0.3761 -0.414 0.2608

43 50.00

46.196 5.9760

6

52.784

9

4.3634

2

48.404

7

3.9660

1

50.181

3

1.4898

1 0.50 0.3532

19

0.2414

05

0.5787

97

0.4774

72

0.5603

55

0.4621

34

0.5104

02

0.1866

9 Est.* = Estimate

It can be clearly observed from the simulation study that the estimates are very close

to the true parameters, and the accuracy is increased as sample size increased.

Therefore, the method of MLE provides precise and accurate estimates of true

parameters for TPF distribution.

5.3.8 Application of transmuted Power function distribution

To investigate the performance of TPF distribution, two real datasets are considered

and described in this section.

Dataset 1: Communication transmitter ARC-1 VHF failure data was collected by the

Mendenhall and Hader (1958). In this data set two separate components confirmed

and unconfirmed failure transmitters were studied. Mendenhall and Hader fitted the

exponential distribution on these data sets. Saleem and Aslam (2010) utilized this data

for the PF distribution after transformed it in the defined range of the distribution.

They explained that the random variable of PF distribution can be generated by an

125

exponential random variable by using the transformation . And the

properties of the data remain same by this transformation. Herein, we also considered

the Mendenhall and Hader confirmed failure data and this transformation to study the

application of the TPF distribution. The method of MLE is employed, and the

parameter estimates and variance-covariance matrix are computed using this dataset.

The results are provided in Table 5.5.

Table 5.5: Estimated parameters with goodness of fit criterion of the TPF

distribution and PF distribution

Model Parameter Estimates Standard Error K-S

TPF

0.00041

0.10838 0.0518

PF 0.00030 0.1150

Exponential 15.5543 0.8698

Pareto 0.02237 0.8211

Log-Logistic

0.1091

10.837 0.9143

On the basis of estimates of TPF distribution hessian matrix and variance and

covariance matrix is computed as respectively

(

) ( )

(

)

Thus the diagonal elements are the variance of estimates, therefore C.I

of the and are estimated as respectively [ ] and

[ ]

Only the PF distribution is the competitor of the TPF distribution from the considered

models. Even then the presentation of PP-plots in Figure 5.12 and the results in Table

5.5 indicates that the TPF distribution is most fitted distribution rather than the PF

distribution.

126

Figure 5.12: PP-plot of TPF distribution and PF distribution for Communication

transmitter failure data

It is already mentioned that the TL-moments provide the more reliable descriptive

statistics of the data. So these moments are calculated and reported in Table 5.6 using

the dataset 1.

Table 5.6: First four C-moments and TL-moments of for Communication transmitter

failure data

r Moments L-moments TL-moments LL-moments LH-moments

1 0.000869 0.000869 4.866×10-6

1.625×10-6

0.001738

2 0.000419 0.000868 4.847×10-6

2.431×10-6

0.001300

3 0.000276 0.000865 5.353×10-6

3.225×10-6

0.001150

4 0.000206 0.000860 5.974×10-6

4.003×10-6

0.001071

Dataset 2: The second application is based on the annual maximum precipitation data

of the Karachi city, Pakistan. The precipitation records are necessary for flood defense

systems, water management studies and prediction of the flood and drought, to

minimize the risk of large hydraulic structures. The considered dataset comprise on 59

annual maximum precipitation records for the years 1950–2009, one of the value for

the year 1987 is missing, rest of the values are used for the analysis.

127

Figure 5.13: Time series plot of annual maximum precipitation data of Karachi,

Pakistan.

To investigate the randomness of the dataset a time series plot and non-parametric

Bartels run test has been applied. The visual inspection and corresponding p-value

(0.7820) of the run test do not support to reject the null hypothesis of randomness. It

seems logical to study this data through probabilistic models. Additionally, Costa,

Fernandes and Naghettini (2015) described that the use of an upper-bounded

distribution for the analysis of the precipitation records is quite handy, based on the

assumption that precipitation records have a physical limit. So, the considered

precipitation dataset in millimeter is modeled by TPF and PF distributions and the

parameters of these densities are estimated by the method of MLE. The estimated

results are listed in Table 5.7.

Table 5.7: Annual maximum precipitation data of Karachi, Pakistan

117.6 157.7 148.6 11.4 5.6 63.6 62.4 11.8 6.5 54.9

39.9 16.8 30.2 38.4 76.9 73.4 85 256.3 24.9 148.6

160.5 131.3 77 155.2 217.2 105.5 166.8 157.9 73.6 291.4

210.3 315.7 107.7 33.3 302.6 159.1 78.7 33.2 52.2 92.7

150.4 43.7 68.3 20.8 179.4 245.7 19.5 30 270.4 160

96.3 185.7 429.3 184.9 262.5 80.6 138.2 28 39.3

Model Estimates SE AIC AICC BIC K-S

TPF 0.79908

0.90374

0.08149

0.09508 691.805 692.016 695.994 0.09026

PF 0.55402 0.07153 703.727 703.796 705.821 0.20262

128

Using the Precipitation data Hessian matrix and variance and covariance matrix is

computed as respectively

*

+ ( )

*

+

Thus the diagonal elements are the variance of estimates, therefore

C.I for the and are estimated as respectively [0.6394, 0.9588] and [0.7173,

1.0000] respectively. To observer the data trend, first four C- and L-,TL-, LL- and

LH-moments are evaluated for the data set 2 and the results are reported in Table 5.8.

Table 5.8: First four C-moments and TL-moments of annual maximum precipitation

data

r Moments L-moments TL-moments LL-moments LH-moments

1 0.005547 0.005547 0.000184 0.000062 0.011032

2 0.002761 0.005485 0.000180 0.000091 0.008136

3 0.001837 0.005363 0.000193 0.000118 0.007032

4 0.001377 0.005185 0.000208 0.000142 0.006339

Figure 5.14: PP-plot of TPF distribution and PF distribution for precipitation data

The comparison of PF and TPF distributions by PP-plots and other goodness of fit

measures indicate that the transmuted distribution provides better results than the

parent distribution.

5.4 Conclusion

In this Chapter, two new double-bounded models are proposed those are the

generalization of the parent distributions. The primary objective to introduce these

129

new models is to provide more flexible environment than the standard one to model

the double-bounded real world situations. To show the flexibility of the new models,

the pdf and cdf are graphed assuming various values of the parameters. With this,

some important statistical properties, densities of the lowest, highest, th order

statistics, the joint density of the two order statistics and generalized TL-moments

with its special cases are derived. The method of MLE is approached for the

estimation of the parameters. The fitting and applicability of the transmuted double-

bounded distributions are exemplified through the real datasets those are taken from

different fields. The appropriateness of the new models is tested through the AIC, KS-

test and LR-test goodness of fit measures. The PP- and Ecdf plots are also sketched to

observe the fitting on the data, graphically. According to all these goodness of fit

criteria, the new double-bounded transmuted distributions have been provided a better

fit than the parent distributions. Therefore, we hope that the new double-bounded

distributions will provide a flexible environment and better fitting on double-bound

data in various disciplines.

130

6. CHAPTER 6

Relations between Transmuted and Parent distribution’s

Moments of Order Statistics

6.1 Introduction

Accurate fitting and modeling of the real dataset using probability distributions have

immense importance in statistics. In this context, transmuted distributions are

considered to be more useful than their parent distributions. The QRTM embeds an

additional parameter in parent density to generate a transmuted density. Due to an

additional parameter, it is complicated and exhaustive to deal directly with the

transmuted distribution, especially for the order statistics analysis. Nevertheless,

various studies have proved the importance of order statistics for proper analysis of

the extreme events. Therefore, to derive the order statistics conveniently, here in this

Chapter, the relationships between transmuted and parent distributions are established

for the single and product moments. In addition, the generalized TL-moment of the

transmuted distribution and its special cases are derived using single moments of the

parent distribution.

Class of transmuted distribution is introduced by Shaw and Buckley (2009). This

class is very useful for modeling a variety of data including extreme value analysis as

it is proved in Chapter 4 and 5. The detail about transmuted distribution is provided in

Section 2.4, for convenience, once again the expression to obtain the cdf of the

transmuted distribution is given below,

(6.1)

which on differentiation yields the corresponding pdf

[ ] (6.2)

where and are the cdf and pdf of the parent distribution respectively.

131

The transmuted density becomes more flexible to model even the complex data as

many researchers proved the versatility and flexibility of the transmuted distributions

(detail provided in Chapter 2).

Balakrishnan and Rao (1998) mentioned that the order statistics are advantageous for

extreme values analysis, TL-moment estimation, in the derivation of the best linear

unbiased estimators and inferential statistics. Therefore, single and product moments

of order statistics and their relationships have achieved considerable attention and

widely studied in the literature, for details we refer to the Arnold and Balakrishnan

(1989). The structure of the probability distributions examined more closely by order

statistics. Nevertheless, the derivation of the moments of order statistics is not a

straight forward task for many distributions, especially for the generalized distribution

such as transmuted class of distribution.

In this study, our primary intention is to reduce the complications those appear in the

derivation of the moments of order statistics for transmuted distributions. Since we

have developed the relationships between transmuted and parent distributions. It will

provide lot of convenience without going into complexities associated with

transmuted distributions. In addition, these relationships are also helpful to obtain TL-

moments of the transmuted distribution.

Rest of the Chapter is organized as follows. In Section 6.2, the relationship between

transmuted and parent distribution has been derived for single moments. The product

moments relationship between the transmuted and parent distribution are established

in Section 6.3. In Section 6.4 the generalized TL-moment of the transmuted

distribution are derived with the help of the single moments of the parent distribution.

The proposed relationships are exemplified theoretically and empirically in Section

6.5 by considering two distributions, Power function, and exponential distribution. In

Section 6.6, we have discussed the parameter estimation of the transmuted densities

through the proposed relationships assuming the concept of MoM by a simulation

study. In Section 6.7, two real-life data sets are used to illustrate the established

relationships and parameter estimation approach. Finally, the study has been

concluded in Section 6.8.

132

6.2 Relation for single moments

Let denote the corresponding order statistics of

independent continuous random variables with ascending order of

magnitude. The is the order statistics and its pdf is given by

Balakrishnan and Cohen (1991) and Arnold et al. (1992), in the following form

[ ] [ ] (6.3)

where [ ] .

The th moment of order statistics for the transmuted distribution is denoted by

and for the parent distribution is denoted by

. These single moments hold the

following relations for and

∑ ∑ (

) (

)

(6.4)

0

1

Proof : From (6.3) we have the following equation for and

∫ [ ] [ ]

Using (6.1) and (6.2), we get

∫ [ [ ] [ ]

[ ]]

expanding binomially, we obtain

∑ ∑ (

) (

)

∫

[ ]

[ ] [ ( )]

∑ ∑ (

) (

)

133

[ ∫

[ ] [ ]

∫

[ ] [ ] ]

simplifying the resulting expression, we get the required relation as given in (6.4).

The pdf of extreme order statistics when and , may be obtained from (6.4)

and can also be derived directly using the following expression, those are defined by

Arnold et al. (1992) as follows

[ ] (6.5)

and

[ ] (6.6)

respectively. The pdf of transmuted distribution for the smallest order statistics when

and is derived as follows

∑ (

)

0

1

and for and is obtained as follows

∑ (

)

0

1

In particular the pdf for and is obtained from (6.3) as

given below

[ ] [ ]

and using this, a relationship of single moments of order statistics is derived as

follows

∑ (

)

0

1

To understand the relationship more simply first few particular single moments

relations between the transmuted and parent distributions are obtained as follows

134

(6.7)

(6.8)

*

+ (6.9)

and

(6.10)

In this way, it is cleared that to calculate the first 3 3 single moments of order

statistics matrix of the transmuted distribution, such as

[

],

we need of 3 6 single moments of order statistics matrix of the parent distribution

[

]

In general to obtain the single moments of order statistics matrix of the

transmuted distribution the single moments of order statistics matrix of the

parent distribution are required.

6.3 Relation for product moments

The joint pdf of and is given by Balakrishnan and Cohen

(1991) and Arnold et al. (1992) as

[ ] [ ] [ ]

(6.11)

where [ ] and and are the cdf

and pdf of the transmuted distribution given (6.1) and (6.2) respectively.

The product moment of transmuted distribution is denoted by

and of parent

distribution is denoted by

. These product moments hold the following relations

for and

∑ ∑ ∑ ∑ (

) (

)

( ) (

)

135

[∑ (

)

∑ (

)

]

(6.12)

Proof. We have the following equation for and from

(6.11) as

∫ ∫ [ ] [ ] [ ]

Using (6.1) and (6.2), we get

∫ ∫ [ ]

[{ } { }]

[ ] [ ]

[ ] ,

expanding binomially, we obtain

∑ ∑ ∑ ∑ (

) (

)

( ) (

)

∫ ∫

[∑ (

) [ ]

[ ] ∑ (

)

[ ] [ ] ]

simplifying the resulting expression, we get the required relation in (6.12). The

particular cases are as follows

for and

136

∑ ∑ ∑ (

)

( ) (

)

[∑ (

)

∑ ( )

]

for and

∑ ∑ ∑ (

)

( ) (

)

[∑ (

)

∑ (

)

]

for and

∑ ∑ (

)

( )

[∑ (

)

∑ ( )

]

and for and

∑ ∑ ( )

(

) [∑ (

)

∑ ( )

]

6.4 Generalized TL-moment of the transmuted distribution

In this section, TL-moments with generalized trimmed for the transmuted distribution

are obtained by the single moments of the parent distribution. In this way, it becomes

more convenient to find TL-moments of the transmuted distributions. The special

137

cases of the generalized TL-moment are also obtained such as TL-moments with the

first trimmed, L-moments without trimmed, LH-moments with trimmed and LL-

moments with trimmed. Elamir and Seheult (2003) established the generalized TL-

moment with smallest and largest trimmed,

as follows

∑ (

)

(6.13)

Where

is a linear function of the expectations of the order statistics. The

generalized TL-moment of the transmuted distribution by the relation of single

moments of the parent distribution is given as follows

∑ ∑ ∑ (

) (

) (

)

0

1 (6.14)

Proof Let and be the cdf and the pdf of the transmuted distribution then the

is defined as

∫ [ ] [ ]

Taking the transmuted distribution in term of parent distribution from (6.1) and (6.2),

and then we get

∫ [ ]

[ ] [ ]

expanding the above expression binomially and after simplification, we have

∑ ∑ (

) (

)

0

1

138

Finally substituted it in (6.13) and resulting got the expression of

that is given

(6.14).

This relationship can be used to find the average value, variance, , and for

the transmuted distribution using the moments of the parent distribution. The special

cases of the generalized TL-moment, L-, TL-, LL- and LH-moments are obtained in

the following form

∑ ∑ ∑ (

) (

) (

)

(6.15)

0

1

∑ ∑ ∑ (

) (

) (

)

0

1 (6.16)

∑ ∑ ∑ (

) (

) (

)

0

1 (6.17)

∑ ∑ ∑ (

) (

) (

)

0

1 (6.18)

respectively.

6.5 Relationship between single and product moments of

transmuted and parent distribution

In this section we exemplify the derived relations of the single and product moment of

order statistics with two densities, Power function and exponential distribution.

139

6.5.1 Moments relations of Power and transmuted Power function distribution

PF distribution is quite popular for modeling lifetime data and especially where

double bounded data is observed. Meniconi and Barry (1996) compared this

distribution with other candidate distributions for the reliability of electrical devices

and found that the fitting of this distribution is better than others. There is a lot of

literature available on the characteristics and estimation of the PF distribution.

Balakrishnan et al. (2000) covered its order statistics aspects comprehensively.

Let again be a random sample of size from the PF distribution with

parameter and , have the following form of the pdf

(6.19)

and with cdf

(6.20)

In the study of the moments of order statistics, we may take the scale parameter

in equation (6.19) and (6.20), without loss of generality. Arnold et al. (1992) and

Balakrishnan et al. (2000) derived the single and product moment for PF distribution

and are given as

⁄

⁄

(6.21)

[ ]

⁄

[ ]

⁄

(6.22)

where (.) is a complete gamma function.

The cdf and pdf of the TPF distribution are obtained by taking (6.19) and (6.20) in

(6.1) and (6.2), respectively

[ ] (6.23)

and

[ ], (6.24)

where is the shape and is the transmuted shape parameter with range .

Some plots of the pdf , given in (6.24) are sketched in Figure 6.1 for the

various choices of the parameters.

140

Figure 6.1: The pdfs of TPF distribution for various choices of the parameters.

The single moments of the transmuted distribution are derived and are listed below

∑ ∑ (

) (

)

(6.25)

[ (

)

⁄

( )

⁄ ]

taking specific values of the r and n in (6.25), we get

(6.26)

[ ]

(6.27)

[ ]

(6.28)

The general expression of the product moment order statistics of the transmuted

distribution is derived as follows

∑ ∑ ∑ ∑ (

) (

) ( ) (

)

141

(

) [∑ (

)

⁄

[ ]

[ ] ∑ (

)

*

+

[ ]

⁄ ] (6.29)

Where ,

and

.

Taking specific values of the and in (6.29) we get

(6.30)

and

[

( ) + (6.31)

where .

The single moments in (6.26), (6.27) and (6.28) and the product moments (6.30) are

obtained directly for TPF distribution. Now the derived relationships, given in (6.4)

and (6.12) is employed to express the moments of the TPF distribution

explicitly in term of the moments of the PF distribution . The first few relation

for single moments are obtained using the specific values in the following form

(

) .

/

142

[ ]

[

]

(

)

0 .

/ .

/1

The product moments given in (6.30) are obtained directly for TPF distribution. Now

the derived relationships, given in (6.12) is assumed to obtain the product moment of

the TPF distribution explicitly in term of the moments of the PF distribution

.

*

,

-+

0 .

/ 2 .

/

.

/ .

/31

[

]

Now we show the single moments relations between the transmuted and parent

distribution empirically. In Table 6.1, the means (single moments with ) of the

143

PF distribution for sample sizes [ ] and of all order statistics are

provided.

Table 6.1: Means of PF distribution of order statistics for [ ]

n r n r n r

1 1 0.3333 0.7500 5 0.5357 0.8882 3 0.1091 0.6520

2 1 0.1667 0.6429 6 0.7500 0.9474 4 0.1818 0.7244

2 0.5000 0.8571 7 1 0.0278 0.4528 5 0.2727 0.7848

3 1 0.1000 0.5786 2 0.0833 0.6037 6 0.3818 0.8371

2 0.3000 0.7714 3 0.1667 0.7043 7 0.5091 0.8836

3 0.6000 0.9000 4 0.2778 0.7826 8 0.6545 0.9257

4 1 0.0667 0.5341 5 0.4167 0.8478 9 0.8182 0.9643

2 0.2000 0.7121 6 0.5833 0.9043 10 1 0.0152 0.4056

3 0.4000 0.8308 7 0.7778 0.9545 2 0.0455 0.5408

4 0.6667 0.9231 8 1 0.0222 0.4347 3 0.0909 0.6310

5 1 0.0476 0.5007 2 0.0667 0.5796 4 0.1515 0.7011

2 0.1429 0.6676 3 0.1333 0.6761 5 0.2273 0.7595

3 0.2857 0.7788 4 0.2222 0.7513 6 0.3182 0.8101

4 0.4762 0.8654 5 0.3333 0.8139 7 0.4242 0.8551

5 0.7143 0.9375 6 0.4667 0.8681 8 0.5455 0.8959

6 1 0.0357 0.4743 7 0.6222 0.9164 9 0.6818 0.9332

2 0.1071 0.6324 8 0.8000 0.9600 10 0.8333 0.9677

3 0.2143 0.7379 9 1 0.0182 0.4191

4 0.3571 0.8198 2 0.0545 0.5589

In Table 6.2, we listed the means of TPF distribution for sample sizes ,

and of all order statistics. The listed values are

obtained directly and also verified by the proved relationship (6.4). For illustration

first two single moments of TPF distribution are obtained by the relationship and

given below

144

It is true for all sample sizes and for different combination of the parameters values.

Similarly it is easy to make the tables for the product moments to prove the

relationship (6.12), numerically.

Table 6.2: Means of TPF distribution of order statistics for [ ]

n r

1 1 0.4500 0.2833 0.7821 0.6750

2 1 0.2763 0.1297 0.6832 0.5625

2 0.6237 0.4370 0.8811 0.7875

3 1 0.1935 0.0737 0.6217 0.5001

2 0.4421 0.2417 0.8061 0.6875

3 0.7145 0.5347 0.9185 0.8374

4 1 0.1457 0.0473 0.5782 0.4584

2 0.3366 0.1526 0.7525 0.6250

3 0.5476 0.3307 0.8598 0.7501

4 0.7701 0.6027 0.9381 0.8665

5 1 0.1151 0.0329 0.5449 0.4279

2 0.2683 0.1050 0.7112 0.5806

3 0.4391 0.2242 0.8144 0.6917

4 0.6200 0.4017 0.8901 0.7890

5 0.8076 0.6529 0.9501 0.8859

6.5.2 Moments relations of exponential and transmuted exponential

distribution

The exponential distribution is one of the most fundamental distribution in statistics.

It is quite handy for modeling data in actuarial, economics, engineering, basic and

social sciences. Especially it is a common model in life-time and survival analysis.

Consequently, there is vast literature available on the theory and applications of the

exponential distribution, for details see Balakrishnan and Basu (1995). The pdf and

cdf of the exponential distribution for random variable are given by

(6.32)

and

(6.33)

respectively. Without loss of generality we can put in (6.32) and (6.33) to

obtain the exponential distribution for the moment of the order statistics. The general

145

expression of the single moments of order statistics for exponential distribution have

been derived by Arnold et al. (1992), as given below

∑

(6.34)

We get some particular single moments from (6.34), as follow

, ⁄ , ⁄ and ⁄

The general expression to find the product order moment of the transmuted

exponential distribution is given below

∫ ∫

(6.35)

Some of the particular product moments of the exponential distribution using (6.35)

are obtained in the following form

, ⁄ , ⁄ , ⁄ and ⁄

The cdf and pdf of the transmuted exponential distributions using (6.32) and (6.33) in

(6.1) and (6.2), with are obtained as

(6.36)

and

[ ] (6.37)

Where is the transmuted shape parameter with defined range

Some plots of the pdf, given in (6.37) are sketched in Figure 6.2 for the

various choices of the parameters.

The single and product moments of order statistics for transmuted exponential

distribution are derived and few moments are given as follows

146

and

Figure 6.2: The pdfs of transmuted exponential distribution for various choices of the

parameter.

Now we calculate the single and product moments by the relationships of the

transmuted and parent distribution those are given in (6.4) and (6.12) and finally we

got the same results as we obtained directly.

(

)

(

) (

) (

)

and

*

{ }+

[

{

}]

147

The table of the single order moments for the exponential distribution has been

provided by Arnold et al. (1992) for [ ] . To provide the empirical

illustration of the single moments relations between transmuted exponential and

exponential distribution, we also present these moments in Table 6.3 for [ ] .

Table 6.3: Means of exponential distribution of order statistics for [ ] .

n r n r n r n r

1 1 1.0000 6 2.4500 5 0.7456 6 0.7365

2 1 0.5000 7 1 0.1429 6 0.9956 7 0.9365

2 1.5000 2 0.3095 7 1.3290 8 1.1865

3 1 0.3333 3 0.5095 8 1.8290 9 1.5199

2 0.8333 4 0.7595 9 2.8290 10 2.0199

3 1.8333 5 1.0929 10 1 0.1000 11 3.0199

4 1 0.2500 6 1.5929 2 0.2111 12 1 0.0833

2 0.5833 7 2.5929 3 0.3361 2 0.1742

3 1.0833 8 1 0.1250 4 0.4790 3 0.2742

4 2.0833 2 0.2679 5 0.6456 4 0.3854

5 1 0.2000 3 0.4345 6 0.8456 5 0.5104

2 0.4500 4 0.6345 7 1.0956 6 0.6532

3 0.7833 5 0.8845 8 1.4290 7 0.8199

4 1.2833 6 1.2179 9 1.9290 8 1.0199

5 2.2833 7 1.7179 10 2.9290 9 1.2699

6 1 0.1667 8 2.7179 11 1 0.0909 10 1.6032

2 0.3667 9 1 0.1111 2 0.1909 11 2.1032

3 0.6167 2 0.2361 3 0.3020 12 3.1032

4 0.9500 3 0.3790 4 0.4270

5 1.4500 4 0.5456 5 0.5699

Table 6.4 presents the single moments of the transmuted exponential distribution for

sample sizes and of all order statistics. These moments

are also checked by the proved identity (6.4) such as

= 0.3542.

148

It is true for all sample sizes and combination of the parameters values. One can also

notice that when , the central moments are exactly equal to the parent

distribution moments which is the basic property of the transmuted distribution.

Table 6.4: Means of transmuted exponential distribution of order statistics for

[ ]

n r n r

1 1 1.2500 1.0000 0.7500 5 1 0.3121 0.2000 0.1365

2 1 0.6875 0.5000 0.3542 2 0.6455 0.4500 0.3123

2 1.8125 1.5000 1.1458 3 1.0450 0.7833 0.5566

3 1 0.4854 0.3333 0.2313 4 1.6001 1.2833 0.9455

2 1.0917 0.8333 0.6000 5 2.6473 2.2833 1.7991

3 2.1729 1.8333 1.4188 6 1 0.2662 0.1667 0.1133

4 1 0.3788 0.2500 0.1717 2 0.5418 0.3667 0.2525

2 0.8053 0.5833 0.4100 3 0.8528 0.6167 0.4320

3 1.3781 1.0833 0.7900 4 1.2373 0.9500 0.6813

4 2.4379 2.0833 1.6283 5 1.7815 1.4500 1.0776

6 2.8205 2.4500 1.9433

6.6 Parameter estimation

In this section, parameters of the considered transmuted distributions are estimated by

combining the identities of the single moments of order statistics and MoM

procedure. These identities established by the Arnold et al. (1992) and we obtained

these identities for both parent and transmuted distribution as follows

∑

(6.38)

and

∑

(6.39)

respectively.

Here we assumed the full transmuted densities without assuming scale parameter

equal to one. Now the TPF distribution and transmuted exponential are two parameter

densities, so it is required to calculate the

and

to derive the parameter

estimators. These moments can also be obtained directly or by the moment

relationships. The identities for the TPF distribution are given as follows

149

(6.40)

and

(6.41)

The simultaneous solution of these equations yields the parameter estimators, but the

exact close form of the estimators is not possible for this distribution. So the estimates

of parameters are obtained by solving the above equations numerically.

The parameter estimator for transmuted exponential distribution can be obtain by

solving the following equations

and the simultaneous solution yields the estimators for the parameter of transmuted

exponential distribution in the following form

.

√ (

)

/ (6.42)

and

.

(

)

√ (

)

/ (6.43)

This parameter estimation approach is investigated through simulation study

assuming three different sample sizes for transmuted exponential distribution. We

generated 10,000 random samples for each sample size to calculate the average

parameter estimates and their mean square errors (MSEs). The results of the

simulation study are reported in Table 6.5. The simulation study is showed that the

performance of the estimations approach is reliable, as the estimates are very close to

true parameter with very low value of MSEs.

150

Table 6.5: The average estimates with their corresponding MSEs with varying sample

size and parameters of the transmuted exponential distribution

n

20 1 -0.5 0.92975 0.0703 -0.44643 0.0536

0.5 0.93159 0.0684 0.46791 0.0321

10 -0.5 9.06104 0.9390 -0.40624 0.0938

0.5 9.59317 0.4068 0.57993 0.0799

50 -0.5 49.0715 0.9285 -0.48868 0.0113

0.5 49.1356 0.8644 0.56079 0.0608

50 1 -0.5 0.96051 0.0395 -0.49057 0.0094

0.5 0.93718 0.0628 0.49771 0.0023

10 -0.5 9.62572 0.3743 -0.54498 0.0450

0.5 9.76254 0.2375 0.46086 0.0391

50 -0.5 49.5385 0.4615 -0.45672 0.0433

0.5 50.8619 0.8619 0.53118 0.0312

100 1 -0.5 0.96051 0.0395 -0.47645 0.0236

0.5 0.98257 0.0174 0.49962 0.0004

10 -0.5 9.80105 0.1990 -0.47285 0.0272

0.5 10.0910 0.0910 0.48541 0.0146

50 -0.5 50.1465 0.1465 -0.52320 0.0232

0.5 50.0254 0.0254 0.50634 0.0063

6.7 Application

In this section, we analyzed the two real life data sets to demonstrate the applicability

of the established results and parameter estimation approach.

6.7.1 Application of the transmuted Power function distribution

The application of the PF distribution is illustrated using data on economic efficiency

scores of the firms. The measurement of economic efficiency was firstly proposed by

Farrell (1957) in terms of technical and allocative efficiency. Technical efficiency

(TE) is attaining higher level of production with given set of inputs. When the

efficiency of any firm is considered with reference to its marginal revenue and cost

then such efficiency is an allocative efficiency. We have used data for 85 firm of

Pakistan, and the TE scores are calculated. These TE scores are independently and

identically distributed.

151

For this data set, we obtained

and

, using these values

we obtained the parameter estimates for TPF distribution and

.

Figure 6.3: Empirical and PP-plot for fitting the TPF distribution

The L-, TL-, LL- and LH-moments for the efficiency score are calculated directly and

using the relationship (6.14). These both methods provided the same results and these

moments are presented in Table 6.6.

Table 6.6: First four L-, TL-, LL- and LH-moments for technical efficiency scores

data

r L-moments TL-moments LL-moments LH-moments

1 0.810197 0.828952 0.726983 0.893410

2 0.083213 0.045101 0.076476 0.048343

3 -0.01875 -0.00688 -0.01786 -0.00713

4 0.008045 0.002264 0.007710 0.002346

6.7.2 Application of the transmuted exponential distribution

In this section, the moments relationships and the method of parameter estimation are

exemplified by the real-life data set using exponential distribution. The data set is

about the mileages at the time of failure of 19 military vehicles. This data set is

already considered by Arnold et al. (1992), Lawless (2011) and many others for the

application of the exponential distribution. The recorded mileage are 162, 200, 271,

302, 393, 508, 539, 629, 706, 777, 884, 1008, 1101, 1182, 1463, 1603, 1984, 2355,

2880.

152

To find the parameter estimates from the estimators are given in (6.40) and (6.41) the

two moments,

and

are required. These moments calculated using the

relationship (6.4) and directly, we obtain

997.2105 and

1543433.

Substituting these moments in the estimators and obtained the estimates, 662.73

and 1.00.

Figure 6.4: Empirical and PP-plot for fitting the transmuted exponential distribution

The moments of the recorded mileage data are calculated and are presented in Table

6.7.

Table 6.7: First four L-, TL-, LL- and LH-moments for military vehicles failure data

r L-moments TL-moments LL-moments LH-moments

1 1380.69 1290.43 960.170 1801.21

2 420.518 214.397 247.699 383.078

3 90.2527 29.1161 18.0420 102.295

4 63.1898 16.1446 21.3006 57.6867

The fitted distribution function on empirical distribution function and PP-plot of the

TPF and exponential distribution are plotted in Figure 6.3 and Figure 6.4. It is noticed

that the transmuted distribution provided the good fit for both data sets.

153

6.8 Concluding remarks

In this Chapter, we have derived the relationships for single and product moments of

order statistics between transmuted and parents distributions. The relationship is also

established to obtain the generalized TL-moment of the transmuted distribution

through single moments of parent distribution. Using this relationship one can easily

derive and compute the L-, TL-, LL- and LH-moments of the transmuted distributions

using the single moments of the parents distributions. Both single and product

moments relationships of order statistics are exemplified theoretically and empirically

assuming exponential and PF distribution. The derived relationships are used to

estimate the parameters of the transmuted distributions and parameter estimation

approach is also investigated through a simulation study. In addition, two real data

sets are used to support our argument that moment relationship between transmuted

and parent distribution is very useful. We hope that the developed relationships will

provide an easy way to analyze and apply the transmuted distribution without

considering some very hard mathematics.

154

7. CHAPTER 7

Summary, Conclusions and Recommendations

Statistics of univariate extremes has been successfully used in various fields, where

interest is in the estimation of magnitude and frequency of extreme or rare events.

Indeed, extreme events can have disastrous consequences on human activities. To

save from potential losses of any future extreme event, a significant development has

been established in Statistics during the last two decades. In this development,

researchers preferably tried to model the extreme events data using probabilistic

distributions, to observe their nature and to predict their forthcoming frequency. As a

continuity of this development, we have also made an effort to contribute to this

dimension, and the main findings and conclusions are summarized as follows.

7.1 Parameter estimation by method of L- and TL-moments

To model the extreme value frequency data and to assess the uncertainty, Singh-

Maddala, Dagum and generalized Power function distributions are suggested in this

study. In extreme value frequency analysis, L-moments and TL-moments play a vital

role, so these moments are derived in generalized form, and particularly first four L-

and TL-moments are obtained for each of the distribution. L- and TL-moments ratio

estimators are also derived for these distributions, as the moments ratios provide

substantial accurate results.

The method of L-moments and method of TL-moments are developed for Singh-

Maddala, Dagum, and generalized Power function distributions to estimate their

parameters. These developed estimation methods are compared with the method of

moments and method of maximum likelihood estimation by employing simulation

and real data studies. The simulation studies indicated that the estimates of the method

of L- and TL-moments tend to produce unbiased results with minimum mean square

155

error than the other considered estimation methods. It is also noticed that the bias and

mean square error are inversely related to the sample size.

To explore the best method of estimation in real data application for Singh-Maddala

distribution monthly maximum temperature of Jacobabad, Pakistan, (from January

1981 to December 2013) has been considered. The parameters are estimated by the

methods of L-moments, method of TL-moments, method of moments and method of

maximum likelihood estimation. The performance of these estimation methods is

compared with Kolmogorov- Smirnov goodness of fit test, PP-plots, and empirical cdf

plots. It has been found that the method of TL-moments produced a more precise

result and provided better fitting than all other considered methods of estimation. The

conventional moments, L-moments and TL-moments and their moment ratios are also

calculated for this data set.

To propose a most appropriate estimation method for Dagum distribution for the

analysis of extremes, the performance of the four considered estimation methods are

compared. This comparison carried out using the annual maximum wind speed at

Vancouver, Canada. The evaluation on the basis of goodness of fit criteria revealed

that the method of TL-moments performed better than all other estimation methods.

The method of L-moment is found the second best option for the parameter estimation

of the Dagum distribution. Similarly, the performance and fitting of the four

considered methods of estimation are also investigated for the generalized Power

function distribution for the analysis of the extreme events. Annual maximum

precipitation data of the Karachi, Pakistan has been considered for real data

application. Here, the performance of the method of L-moments observed better than

the rest of the considered estimation methods. Generally, we can conclude that

method of TL-moment and method of L-moments are the most reliable estimation

methods for the fitting of the considered probability densities on the extreme events

data. L- and TL-moments are also provided good summarization and description of

the observed data.

7.2 New proposed Transmuted Distributions for Extreme Value

Analysis

The quality of the statistical analysis for extreme value data heavily depends on the

assumed probability distribution. Therefore, it is a common practice that some

156

suitable candidate distributions compared and reviewed to take the decision regarding

truly fitted distribution. In fact, the use of probability distributions has several

advantages. Such as, probability distribution presents a smooth and consistent

interpretation of the data, provides more accurate statistical information, including,

quantiles and more realistic range of the random variable that it can assume. To obtain

these advantages in extreme value analysis, some new flexible distributions are

proposed in this dissertation, using quadratic rank transmutation map to model the

variety of data. The quadratic rank transmutation map provides a new generalization

of any distribution. Here, the proposed distributions are transmuted Singh-Maddala,

transmuted Dagum and transmuted New distribution.

The first proposed distribution, the transmuted Singh-Maddala distribution, as a

generalization of the parent (Singh-Maddala) distribution. This distribution is quite

flexible, and its application diversities increased due to the additional transmuted

parameter. To show the flexibility of new density the plots of the pdf, cdf, reliability

function and hazard functions have been sketched. This graphical presentation

justified the flexibility and versatility of the transmuted Singh-Maddala distribution.

The moments and other basic properties of the proposed distributions have been

derived. The densities of the lowest, highest, th order statistics, the joint density of

the two order statistics and generalized TL-moments are also studied. The parameter

estimation is obtained by the maximum likelihood estimation via Newton-Raphson

approach. To evaluate its fitness, five goodness of fit criterion are considered for the

selection of most appropriate model assuming a real data set. In real data set, monthly

household expenditure data set has been considered and this data taken from

Household Integrated Economic Survey, Pakistan Bureau of Statistics. The results of

AIC, AICC and BIC indicate that transmuted Singh-Maddala distribution provides a

better fit than the parent distribution. The likelihood ratio test is also employed and

the results justify the importance of the additional transmuted parameter. The fitted

densities are compared with the empirical histogram and empirical cdf. The graphical

presentation has been showed that the transmuted Singh-Maddala distribution

followed the empirical data pattern better than the parent distribution for the observed

expenditure data set.

Transmuted Dagum distribution is the second proposed distribution. Aforementioned

statistical properties, order statistics, and generalized TL-moments are derived for this

157

distribution. The hazard rate function of the transmuted Dagum distribution is

attractively flexible. Therefore, it is useful and suitable for many real life situations. It

showed various shapes like decreasing, increasing-decreasing, bathtub and upside

down bathtub for specific values of the parameters. Parameter estimation of the

transmuted Dagum distribution is approached by the method of maximum likelihood

estimation. The performance of the transmuted Dagum distribution compared with

Dagum distribution by considering the monthly maximum precipitation data of

Islamabad, Pakistan. The proposed transmuted Dagum distribution fitted well on the

data as compared to the Dagum distribution according to all goodness of fit criteria,

such as , AIC, AICC, BIC, Kolmogorov- Smirnov-test and Likelihood-Ratio

test. Additionally, both empirical cdf and PP-plots indicated that the transmuted

Dagum distribution exactly follows the pattern of the empirical data set. These all

results supported us to conclude that the transmuted Dagum distribution is better

model than the parent model for fitting on such data.

The third proposed distribution is the transmuted New distribution. This distribution

exhibits various shapes like as exponentially decreasing and increasing-decreasing

with a heavy tail. Therefore, it is a good option for modeling the extreme events and

lifetime data sets. The reliability analysis, moments, and order statistics have been

explored for transmuted New distribution. In real data application, two data sets have

been considered to prove the superiority and applicability of the proposed transmuted

model. The first data set consisted of 76 observations of the life of fatigue fracture of

Kevlar 373/epoxy that are subject to constant pressure at the 90% stress level until all

had failed. The second data set is about the maintenance actions for the number 4

diesel engine of the U.S.S. Grampus. On the basis of all considered criteria (AIC,

AICC, BIC, Kolmogorov-Smirnov test, PP-plot and empirical cdf plot) the results of

real life data sets showed that transmuted New distribution is superior to the base

distribution.

Finally, we hope that the proposed probability distributions, transmuted Singh-

Maddala, transmuted Dagum and transmuted New distribution will prove to be very

useful in modeling income distribution, actuarial, meteorological and survival

extreme value data analysis.

158

7.3 Double Bounded Transmuted Distributions for Extreme Value

Analysis

In many situations, bounded data sets are observed where tasks are performed

periodically and must be completed before an end-to-end deadline. To model the data

sets those appear from such situations, two transmuted bounded distributions are

developed and proposed in this dissertation. The main motivation for these new

models is to provide a more flexible environment than the standard one in modeling

for double bounded real world situations.

The first proposed continuous double-bounded distribution is the transmuted

Kumaraswamy distribution. This proposed distribution is very flexible with a wide

variety of shapes within its range as the plots of the pdf and cdf showed. Some basic

properties, reliability function, hazard function, densities of the lowest, highest, th

order statistics, the joint density of the two order statistics and generalized TL-

moment with its special cases are derived. Three real situations empirical studies have

been presented to compare transmuted and parent distribution. In first study, the

Muslim population percentage in different countries has been taken based on 2004

Census projection. The data about annual maximum peak flows at the Kalabagh site at

Indus River, Pakistan is used as second application. The third data set was the daily

ozone level measurement in New York, USA. The parameters of transmuted

Kumaraswamy distribution have been estimated for all three data sets by method of

maximum likelihood estimation. The appropriateness of the proposed model has been

tested by AIC, Kolmogorov-Smirnov test, Likelihood-Ratio test, PP-plot and

empirical cdf and compared with its parent models. According to all these goodness

of fit criteria the new double-bounded transmuted distribution provides a better fit

than the parent distribution.

Transmuted Power function distribution is the second proposed double bounded

distribution. All the above mentioned statistical properties, order statistics, and

generalized TL-moments are also derived for this distribution. The hazard rate

function of transmuted Power function distribution has the increasing and bathtub

shaped, which is attractive for the medical, engineering and other disciplines for

survival or reliability analysis. The simulation study and real data applications proved

that the transmuted Power function distribution performs better than the parent

distribution. For modeling civil engineering and hydrology data transmuted Power

159

function distribution is fairly appropriate because of its flexibility, simplicity, and

applicability. It is also useful to estimate the reliability of the systems and dams

infrastructure.

Therefore, the proposed double-bounded transmuted distributions would provide a

flexible environment and better fitting of double-bound data in various disciplines.

7.4 Relationships between Transmuted and Parent distributions

It is found that transmuted distributions are more flexible and useful than their parent

distribution, as it is revealed from above discussion. Transmuted distribution embeds

an additional parameter in parent density to generate transmuted density. To deal

directly with the transmuted density is complicated and exhaustive, especially for

order statistics analysis and for the densities that have more than two parameters. In

this study, relationships between transmuted and parent distributions are established

for the single and product moments of order statistics. These relationships are verified

both theoretically and numerically. In addition, the relation to obtaining the

generalized TL-moment of the transmuted distribution through single moments of

parent distribution is also established. Now using this relation one can easily derive

and compute the L-, TL-, LL- and LH-moments of the transmuted distributions taking

the single moments of the parent distribution. Both single and product moments

relationships of order statistics are exemplified theoretically and empirically assuming

exponential and Power function distribution. The derived relations have been used to

estimate the parameters of the transmuted distributions and parameter estimation

approached also investigated through a simulation study. In addition, we have used

two real data sets to support our argument that moment relationship between

transmuted and parent distribution is very useful. We hope that the developed

relations will provide an easy way to analyze and apply the transmuted distribution

without taking hard mathematics.

7.5 Recommendations

The following important and fundamental developments and directions are

recommended for the future research in this area.

i) This dissertation introduces the approach of L and TL-moments for parameter

estimation of some specified distribution and analysis of the extreme value

160

data related to these distributions. The approach can be utilized for some new

classes of distributions such as Transmuted, Beta, Exponentiated,

Kumaraswamy, McDonald etc.

ii) Some heavy-tailed distributions are transmuted for better analysis of the

extreme value data. There are many other heavy tail distributions available in

the literature that can be transmuted to get the good fit on the data on extreme

events.

iii) Two double-bounded distributions are transmuted here for the better analysis

of the extreme value data. There are many other double-bounded distributions

present in the literature that can be generalized by quadratic rank

transmutation map to model the data of extreme events.

161

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