characterization and estimation of weighted …

CHARACTERIZATION ANDESTIMATION OF WEIGHTED

PROBABILITY DISTRIBUTIONS

THESIS

SUBMITTED FOR THE AWARD OF THE DEGREE OF

Doctorof PhilosophyIN

STATISTICS

BY

AIJAZ AHMAD DAR

UNDER THE SUPERVISION OF

Prof.Aquil Ahmed(Supervisor)

DEPARTMENT OF STATISTICS AND OPERATIONS RESEARCHALIGARH MUSLIM UNIVERSITY

ALIGARH-202002, INDIA

2019

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DEPARTMENT OF STATISTICS & OPERATIONS RESEARCH

ALIGARH MUSLIM UNIVERSITY

ALIGARH – 202002, INDIA

Candidate’s Declaration

I, Aijaz Ahmad Dar, Department of Statistics and Operations Research

certify that the work embodied in this Ph.D. thesis is my own bonafide work

carried out by me under the supervision of Prof Aquil Ahmed at Aligarh Muslim

University, Aligarh. The matter embodied in this Ph.D. thesis has not been

submitted for the award of any other degree.

I declare that I have faithfully acknowledged, given credit to and referred

to the research workers wherever their works have been cited in the text and the

body of the thesis. I further certify that I have not willfully lifted up some

other's work, para, text, data, result etc. reported in the journals, books,

magazines, reports, dissertations, thesis, etc. or available at web-sites and

included them in this Ph.D. thesis and cited as my own work.

Dated: ……………… (Aijaz Ahmad Dar)

……………………………………………………………………………………

Certificate from the Supervisor

This is to certify that the above statement made by the candidate is correct

to the best of my knowledge and belief.

Signature of the Supervisor : ......................................................

Name and Designation : Professor Aquil Ahmed

Department: Statistics & Operations Research, Aligarh Muslim

University, Aligarh.

(Signature of the Chairman of the Department with seal)

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Phone: +91 571 2701251 (Office), e-mail: [email protected]




Athar Ali Khan

Professor & Chairman

Course/Comprehensive Examination/Pre-submission

Seminar Completion Certificate

This is to certify that Mr. Aijaz Ahmad Dar, Department of Statistics &

Operations Research has satisfactorily completed the course work/

comprehensive examination and pre-submission seminar requirement which is

part of his Ph.D. (Statistics) programme.

Dated: (Prof. Athar Ali Khan)

Chairman

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Copyright Transfer Certificate

Title of the Thesis : CHARACTERIZATION AND ESTIMATION OF

WEIGHTED PROBABILITY DISTRIBUTIONS

Candidate’s Name : AIJAZ AHMAD DAR

Copyright Transfer

The undersigned hereby assigns to the Aligarh Muslim University, Aligarh

copyright that may exist in and for the above thesis submitted for the award of

the Ph.D. degree.

(Aijaz Ahmad Dar)

Signature of the Candidate

Note: However, the author may reproduce or authorize others to reproduce

material extracted verbatim from the thesis or derivative of the thesis for

author’s personal use provided that the source and the University’s

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PrefaceThis thesis entitled "Characterization and Estimation of Weighted Probability distribu-tions" comprises of five chapters each consisting several sections and subsections. Thechapter wise content of the thesis is summarized as:

Chapter 1 is devoted to the introduction and presents all the sufficient and relevantinformation regarding the subject matter used in the subsequent chapters in detail. Thischapter which is imperative for understanding the subject matter, comprises of generalintroduction about the concept of weighted distributions, their importance, literaturereview related to different statistical measures/techniques used and objectives of thestudy.

Chapter 2 In this chapter, we have introduced the weighted version of Maxwell-Boltzmann distribution on considering the weight function w(x) = xω . The introducedversion is named as weighted Maxwell-Boltzmann distribution and abbreviated asWMD. Expressions related to different statistical measures and functions associatedwith WMD like descriptive statistics, information measures and income distributioncurves have been derived and studied in detail. Inverse sampling method is used togenerate random numbers from WMD. Parameters of the distribution are estimated byusing the procedure of maximum likelihood estimation and method of moments. Toshow the validity and adequacy of WMD in statistical modeling, it is fitted to fourdifferent types of data sets along with its special cases and a comparison is made interms of distribution of best fit for the considered data sets. Akaike information crite-rion (AIC), Akaike information criterion corrected (AICc) and Bayesian informationcriterion (BIC) are used as tools for finding out the best fitted distribution.

Chapter 3 deals with the study of four parameter extension of power function dis-tribution, termed as weighted transmuted power function distribution and abbreviatedas WTPFD. Various properties of the introduced extension are studied and investigated.Cases under which WTPFD can be reduced to mixture of two power function distri-butions are observed. Increasing and decreasing behavior of the density of WTPFDis studied. Nature of the hazard rate and mean residual life function is also observed.Random numbers from the introduced extension are generated by using inverse sam-pling procedure. Parameters are estimated by using the maximum likelihood estima-tion method. Comparison is made between special cases of WTPFD in terms of modelof best fit by making the use of comparison tools, viz., AIC and Kolmogorov-Smirnovtest. To show the importance of WTPFD and to support its validity in describing arandom phenomena, it is fitted to four different types of data sets and a statement ismade that mostly newly developed versions prove to be more efficient and adequate incomparison to the baseline distribution, i.e., power function distribution.

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Chapter 4 Transmuted weighted exponential distribution (TWED) has been intro-duced in Chapter 4. This chapter deals with the development of three parameter ex-tension of exponential distribution by using the concept of transmutation and weighteddistribution. Weight function w(x) = xω is used to derive the weighted version of oneparameter exponential distribution and the resulting version is transmuted by usingthe quadratic rank transmutation map (QRTM). Various properties of TWED are in-vestigated and studied in detail. Expressions for the rth moment about origin, m.g.f.,characteristic function, Renyi entropy, Bonferroni curve and Lorenz curve have alsobeen derived. In addition to it, we have also obtained expressions for the reliabilityfunction, hazard rate, mean residual life, densities of first order, nth order and rth orderstatistics. For illustrating the application of TWED, it is fitted to two types of data setsand distribution of best fit is found among the special cases of TWED by using modelselection tools, viz., AIC, AICc and BIC.

Chapter 5 is devoted to the study of weighted gamma-Pareto distribution. In thischapter, gamma-Pareto distribution (GPD) which belongs to T-X family is consideredand its weighted version is introduced. The introduced version is named as weightedgamma-Pareto distribution and abbreviated as WGPD. Different properties associatedwith WGPD have been studied in detail. GPD is form-invariant under weight functionw(x) = xω . Therefore, a new weight function which is defined as w(x) = 1− (θ/x)λω

is used for the derivation of WGPD. It has been shown that parent distribution (GPD)is smaller than the introduced weighted version (WGPD) in terms of stochastic, fail-ure rate, likelihood ratio and mean residual life ordering. Interestingly, the derivedweighted version of GPD proved to be the generalization of some well known distri-butions, viz., gamma-Pareto, weighted gamma, weighted exponential, generalized ex-ponential, weighted exponential-Pareto, exponential, Pareto and gamma distribution.Random numbers from the introduced version are generated by employing the rejec-tion method. For illustration purpose, WGPD is fitted to a simulated and a real lifedata set in order to show its importance and validity in describing a random process.

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AcknowledgementsFirst and foremost, praises and thanks to the Almighty ALLAH, the merciful and om-niscient, for His immense blessings, He showered on me and for providing me thehealth and capability to move forward successfully.

I would like to express my deep and sincere gratitude to my beloved supervisorProfessor Aquil Ahmed for the most beautiful thing one would like to have that ishis affectionate attitude, support and parentally advice. I am very much obliged tohim for his valuable suggestions, consistent guidance and timely encouragement thatconverted my dream into reality. I would also like to thank him for giving his valuabletime whenever I needed. His dynamism, vision, sincerity and motivation have deeplyinspired me and showed me the path of inculcating the honest, sincere, polite andhelpful behavior. For me, it was and will be a great privilege and honor to work andstudy under his guidance.

I would like to thank Prof. Athar Ali Khan, Chairman, Department of Statistics& Operations Research, Aligarh Muslim University for providing all the necessaryfacilities one would like to have for carrying out the research work, may it be seminarlibrary, computer lab or overall sophisticated environment. I sincerely thank to Prof.Mohammad Masood Khalid, Prof. Qazi Mazahar Ali and Arif-Ul-Islam for the same.

I am willing to express my appreciation and regard to all the teachers who camein my life and contributed to provide me such a beautiful life which is full of respect,dignity and morality. I would also like to thank them for their moral support and wholehearted teaching and guidance. I would like to thank all the non-teaching staff of thedepartment of Statistics & Operations Research, A.M.U., especially to Mr. JabbarAnwar, Mr. Rozer Mahmood, Mr. Zulfiqar Ali, Mr. Arif Uzzaman Khan, Mr. MohdHaroon and Mr. Salman Husain for their entire support.

It is my privilege to express my sincere regards to Dr. Javaid Ahmad Reshi forhis valuable inputs, able guidance, encouragement and whole hearted cooperationthroughout the duration of my research. I would also like to thank him for his friend-ship, empathy and polite nature.

I would like to pray for the noble soul, Sir Syed Ahmad Khan whose endless ef-forts and sleepless nights have provided us such a beautiful environment that one cantransform his dream into reality.

The most overwhelming key behind a child’s success is the positive involvementof his parents so their noble sacrifices can’t be compensated by mere words. I amextremely grateful to my beloved parents for their love, prayers, caring and sacrificesfor educating and preparing me for my future. I would like to thank them for theirsupport and being ready to sacrifice their everything for the sake of my progress. I

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http://www.profaquil.org

fall short of words to thank my sisters Mrs. Shameema Akhter, Ms. Shagufta Akhter,Ms. Shaista Jan and brothers Mr. Mushtaq Ahmad, Mr. Reyaz Ahmad and Mr. SajadAhmad who have always been my strength, encouragement and motivation throughoutmy life. I would also like to thank all my nears and dears for the same.

I am particularly grateful to my senior research colleagues, especially to Mr. FaizNoor Khan Yusufi, Mr. Srikant Gupta, Mr. Ateebur Rehman Sherwani, Mr. MohdArif, Mr. Gulzar-Ul-Hassan, Mr. Akram Raza Khan, Mr. Firdoos Yusuf, Mr. ShowketAhmad lone, Mr. Ahmedur Rehman, Mr. Mehfooz Alam, Mr. Murshid Kamal, Mr.Intekhab Alam, Mr. Najrullah Khan, Ms. Zoha Qayoom, Ms. Afrah Hafeez, Mrs.Sadia Parveen, Mrs. Arti Sharma, Mrs. Haneefa Kauser for their immense support andlove.

I would also like to thank Ms. Bushra Khatoon, Mrs. Saima Zarrin, Mr. Md.Ashraf-Ul-Alam, Mr. Umar Muhammad Modibbo, Mr. Abdul Nasir, Mr. Firoz Ah-mad, Mr. Ahtesham-Ul-Haq, Ms. Bavita Singh, Mrs. Sana, Ms. Shazia Farhin andMs. Nancy Khandelwal for motivating and supporting me.

My friends Mr. Ryhan Abdullah, Mr. Jibraan Fazili, Mr. Syed Aabid Hussain, Mr.Syed Aqib Jalil, Mr. Mohd. Aasim Nomani, Mr. Qazi Azhad Jamal, Mr. Yasir AhmadRaina, Mr. Waseem Ahmad, Mr. Mohd. Amir Ansari, Mr. Syed Mohd. Muneeb, Ms.Neetu Gupta, Ms. Nausheen Hashmi, Ms. Zainab Asim, Ms. Aisha Bhat, Mr. OmerAbdalghani, Mr. Mohammad Abu Jarad, Mr. Abdul Roauf Rather, Mr. Qisar FarooqDar, Mr. Mohd. Shadab, Mr. Mohd. Khalid, Mr. Tauseef Ahmad Naqishbandi, Mr.Amir Ahmad Dar and Mr. Shahid Ahmad do deserve a special mention for their lovelycompany and for believing in me. I am very much thankful to my roommates, Mr.Mohd Iqbal Lone, Mr. Ashiq Hussain Mir and Mr. Shariq for making me feel as if Iwas not away from my family during my stay in RAS-08, S.Z. Hall, A.M.U., Aligarh.I would like to thank my childhood friends Mr. Mansoor Ahmad, Mr. Zahoor Ahmad,Mr. Shakeel Ahmad, Mr. Altaf Ahmad, Mr. Mudasir Ahmad and Mr. Fayaz Ahmadfor believing in me and for their support through the ups and downs of my life.. . .

Aligarh: Aijaz Ahmad Dar

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Contents

List of Figures xvii

List of Tables xix

List of Abbreviations xxi

1 Introduction 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Basic preliminaries and fundamental concepts . . . . . . . . . . . . . 2

1.2.1 Some continuous distributions . . . . . . . . . . . . . . . . . 21.2.1.1 Gamma distribution . . . . . . . . . . . . . . . . . 21.2.1.2 Exponential distribution . . . . . . . . . . . . . . . 31.2.1.3 Maxwell distribution . . . . . . . . . . . . . . . . . 41.2.1.4 Power function distribution . . . . . . . . . . . . . 51.2.1.5 Pareto distribution . . . . . . . . . . . . . . . . . . 5

1.2.2 Approaches used for introducing a new distribution . . . . . . 51.2.2.1 Transmuted family of distributions . . . . . . . . . 6

1.2.2.1.1 Sample transmutation mapping. . . . . . 71.2.2.1.2 Rank transmutation mapping. . . . . . . . 71.2.2.1.3 Quadratic rank transmutation mapping. . . 8

1.2.2.2 Transformed-Transformer (T-X) family of distribu-tions . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Basic reliability concepts . . . . . . . . . . . . . . . . . . . . 101.2.3.1 Reliability function . . . . . . . . . . . . . . . . . 101.2.3.2 Probability density function . . . . . . . . . . . . . 101.2.3.3 Hazard rate . . . . . . . . . . . . . . . . . . . . . . 101.2.3.4 Cumulative hazard rate . . . . . . . . . . . . . . . 10

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1.2.3.5 Mean residual life . . . . . . . . . . . . . . . . . . 101.2.4 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.5 Measures of inequality . . . . . . . . . . . . . . . . . . . . . 12

1.2.5.1 Lorenz curve and Bonferroni curve . . . . . . . . . 131.2.5.2 Gini index . . . . . . . . . . . . . . . . . . . . . . 14

1.2.6 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.6.1 Renyi entropy . . . . . . . . . . . . . . . . . . . . 151.2.6.2 Shannon entropy . . . . . . . . . . . . . . . . . . . 15

1.2.7 Simulation techniques . . . . . . . . . . . . . . . . . . . . . 151.2.7.1 Inverse sampling method . . . . . . . . . . . . . . 161.2.7.2 Rejection method . . . . . . . . . . . . . . . . . . 16

1.2.8 Integration using Monte Carlo simulation . . . . . . . . . . . 181.2.9 Partial ordering . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.10 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.10.1 Point estimation . . . . . . . . . . . . . . . . . . . 191.2.10.1.1 Method of Maximum likelihood estimation: 191.2.10.1.2 Method of moments: . . . . . . . . . . . 20

1.2.10.2 Interval estimation . . . . . . . . . . . . . . . . . . 201.2.10.2.1 Interval estimation using the asymptotic nor-

mal property of M.L.E. . . . . . . . . . . 201.2.11 Comparison criteria and goodness of fit . . . . . . . . . . . . 21

1.2.11.1 Akaike information criterion . . . . . . . . . . . . 211.2.11.2 Bayesian information criterion . . . . . . . . . . . 221.2.11.3 Kolmogorov Smirnov test . . . . . . . . . . . . . . 22

1.3 Weighted distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 241.3.1 Connection between Bayesian inference and theory of

weighted distribution . . . . . . . . . . . . . . . . . . . . . . 261.3.2 Some remarkable theorems related to weighted distributions . 27

1.4 Importance of the study . . . . . . . . . . . . . . . . . . . . . . . . . 271.5 Review of literature . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.6 Objectives of the study . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Weighted Maxwell-Boltzmann Distribution 352.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.1 Derivation of weighted Maxwell-Boltzmann distribution . . . 362.2 Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . 382.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.4 Bonferroni and Lorenz curve . . . . . . . . . . . . . . . . . . . . . . 42

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2.5 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.6 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 Maximum likelihood estimation . . . . . . . . . . . . . . . . 432.6.2 Method of moments . . . . . . . . . . . . . . . . . . . . . . 45

2.7 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7.1 Real life data . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Weighted Transmuted Power Function Distribution 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.1 Derivation of weighted transmuted power function distribution 543.2 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Reliability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5 Entropy measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.7 Bonferroni and Lorenz curve . . . . . . . . . . . . . . . . . . . . . . 663.8 Random number generation . . . . . . . . . . . . . . . . . . . . . . . 673.9 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . 673.10 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.10.1 Real life data . . . . . . . . . . . . . . . . . . . . . . . . . . 693.10.2 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Transmuted Weighted Exponential Distribution 734.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1.1 Derivation of transmuted weighted exponential distribution . . 744.2 Structural properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Reliability measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.4 Renyi entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.5 Order statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.6 Bonferroni and Lorenz curve . . . . . . . . . . . . . . . . . . . . . . 814.7 Random number generation . . . . . . . . . . . . . . . . . . . . . . . 834.8 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . 834.9 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.9.1 Real life data . . . . . . . . . . . . . . . . . . . . . . . . . . 844.9.2 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . 85

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4.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5 Weighted Gamma-Pareto Distribution 895.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.1.1 Derivation of weighted gamma-Pareto Distribution . . . . . . 905.1.1.1 Weight function . . . . . . . . . . . . . . . . . . . 91

5.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.3 Entropy measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4 Random number generation . . . . . . . . . . . . . . . . . . . . . . . 975.5 Maximum likelihood estimation . . . . . . . . . . . . . . . . . . . . 995.6 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.6.1 Simulated data . . . . . . . . . . . . . . . . . . . . . . . . . 1015.6.2 Real life data . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Bibliography 107

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List of Figures

1.1 Relation of gamma distribution with other distributions. . . . . . . . . 31.2 Relation of exponential distribution with other distributions. . . . . . 41.4 Graphical representation of one sample and two sample KS-test statis-

tic D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.1 Density, c.d.f., reliability and hazard rate curves at different values ofω and θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Asymptotic normal curves of M.L.E.’s. . . . . . . . . . . . . . . . . . 502.3 Empirical density and distribution curves along with the fitted ones. . 51

3.1 P.d.f. and c.d.f. curves of WTPFD at different values of θ , α, β and ω . 553.2 Density plots at different values of β . . . . . . . . . . . . . . . . . . 593.3 Hazard rate and m.r.l. at different values of θ ,α,β and ω . . . . . . . 633.4 Density of rth order statistics at θ = 10,α = 5,β = 0.5,ω = 2 and n= 11. 663.5 Gradient of log likelihood function w.r.t. θ at different values of α,β

and ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.6 Empirical along with fitted density and distribution curves. . . . . . . 71

4.1 P.d.f. and c.d.f. plots at different values of ω, λ and β . . . . . . . . . 754.2 3D plot of hyper-geometric function at different values of r and ω . . . 764.3 Hazard rate and mean residual life function at different values of ω, λ

and β . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.4 Empirical density and distribution curves along with the fitted ones. . 86

5.1 P.d.f., c.d.f., reliability and hazard rate curves at different values ofω,α,λ and θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.2 Ratio of target to proposal density at ω = 0.5,α = 5,λ = 5,θ = 10,θ1 =

10,k = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

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5.3 Empirical density and distribution curves along with the fitted ones. . 104

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List of Tables

1.1 W (.) functions suggested by Alzaatreh, Lee, and Famoye (2013) forthe support T ∈ [a,∞) and T ∈ (−∞,∞) . . . . . . . . . . . . . . . . 9

1.2 Some weight functions . . . . . . . . . . . . . . . . . . . . . . . . . 251.3 Different characteristics of gamma, exponential, Maxwell, power and

Pareto distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1 Special cases of WMD at different values of ω . . . . . . . . . . . . . 382.2 Characteristics of WMD at different values of θ and ω . . . . . . . . . 422.3 M.L.E.’s, moment estimates, AIC, BIC and AICc. . . . . . . . . . . . 482.4 95% confidence interval for M.L.E.’s of WMD along with the Fisher

Information and covariance matrix. . . . . . . . . . . . . . . . . . . . 49

3.1 Special Cases of WTPFD . . . . . . . . . . . . . . . . . . . . . . . 563.2 Characteristics of WTPFD at different values of parameters. . . . . . 643.3 M.L.E.’s, AIC and KS-test statistic. . . . . . . . . . . . . . . . . . . . 70

4.1 Some special cases of TWED. . . . . . . . . . . . . . . . . . . . . . 744.2 Characteristics of TWED at different value of ω, λ and β . . . . . . . 774.3 M.L.E.’s and different comparison criteria. . . . . . . . . . . . . . . . 85

5.1 Characteristic of WGPD at different values of ω,α,λ and θ . . . . . . 965.2 M.L.E.’s, -log likelihood and AIC . . . . . . . . . . . . . . . . . . . 1035.3 Special cases of WGPD . . . . . . . . . . . . . . . . . . . . . . . . . 105

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List of Abbreviations

ABED Area Biased Exponential Distribution.ABMD Area Biased Maxwell Distribution.ABPFD Area Biased Power Function Distribution.ABTPFD Area Biased Transmuted Power Function Distribution.ABTUD Area Biased Transmuted Uniform Distribution.ABUD Area Biased Uniform Distribution.AIC Akaike Information Criterion.AICc Akaike Information Criterion Corrected.BIC Bayesian Information Criterion.c.d.f. Comulative distribution function.CV Coefficient of Variation.DMRL Decreasing Mean Residual Life.ED Exponential Distribution.GD Gamma Distribution.GED Generalized Exponential Distribution.GPD Gamma-Pareto Distribution.IFR Increasing Failure Rate.LBED Length Biased Exponential Distribution.LBMD Length Biased Maxwell Distribution.LBPFD Length Biased Power Function Distribution.LBTPFD Length Biased Transmuted Power Function Distribution.LBTUD Length Biased Transmuted Uniform Distribution.LBUD Length Biased Uniform Distribution.MD Maxwell Distribution.M.L.E. Maximum Likelihood Estimator.m.r.l. Mean residual life.MVUE Minimum variance unbiased estimator.NBUE Never Better than Used in Expectation.p.d.f. Probability density function.PFD Power Function Distribution.QRTM Quadratic Rank Transmutation Mapping.RTM Rank Transmutation Mapping.STM Sample Transmutation Mapping.TABED Transmuted Area Biased Exponential Distribution.

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TED Transmuted Exponential Distribution.TLBED Transmuted Length Biased Exponential Distribution.TUD Transmuted Uniform Distribution.TWED Transmuted Weighted Exponential Distribution.UD Uniform Distribution.WED Weighted Exponential Distribution.WEPD Weighted Exponential-Pareto Distribution.WGD Weighted Gamma Distribution.WGPD Weighted Gamma-Pareto Distribution.WMD Weighted Maxwell Distribution.WPFD Weighted Power Function Distribution.WTPFD Weighted Transmuted Power Function Distribution.WTUD Weighted Transmuted Uniform Distribution.WUD Weighted Uniform Distribution.

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Dedicated to my beloved parents and my family. . .

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Chapter 1Introduction

1.1 Introduction

Statistics is the science of drawing inferences about random phenomena in whichchance plays a major role. Being a statistician, the first and foremost concern is topredict some future event with much higher accuracy and the degree of accuracy canbe guaranteed only by using the adequate and suitable models for modeling. In thecurrent era, the applicability of statistics, particularly statistical modeling is so vastthat there is merely any field where statistics does not play any role. Use of statisticsas a whole and statistical modeling in particular, in various disciplines is meaningfulonly if it is used properly. Different statistical investigations give birth to different datasets and problems which leads to the requirement of some new, adequate and flexibledistributions for modeling purpose. Since, there are many existing distributions in thestatistical literature that can be used for modeling a particular data set. These mod-els may not be enough to express the statistical behavior of all the real life randomphenomena, due to which the demand for a relatively larger family of probability dis-tributions arise. In order to overcome such a demand, researchers use different modelconstruction techniques, e.g., the concept of different G-distributions, transmutation,truncation etc. Applications of statistical distributions in describing a random processis so vast that their theory is being widely studied and new distributions come intoexistence. Due to the interest in developing new families of probability models, manygeneralized classes of distributions have been introduced. One of the common featuresof these generalized classes of distributions is that they usually have more parameters,while it is sufficient to use four-parameter distributions for most practical purposes,see Johnson, Kotz, and Balakrishnan (1994). However, at least three or at most fourparameters are needed and the noticeable improvement on using the distribution withmore than four parameters is considered to be ambiguous. Generally, different types

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Chapter 1. Introduction

of G-distributions are used in order to arrive with a new version of an already existingprobability distribution. There is a lot of well known approaches that exist in statisticalliterature and are used to introduce a new extension of some already existing distri-bution. The basic structure of some of these approaches is discussed in the upcomingSection 1.2.2

Keeping in view the interest to add few more probability models to the alreadyexisting, some approaches discussed in Section 1.2.2 and the concept of weighed dis-tributions discussed in Section 1.3, have been used in the subsequent chapters. Since,the main theme of this work is based on the concept of weighted distribution therefore,it is imperative to underline some of the basic concepts related to a probability modelbefore we discuss the concept, genesis and importance of weighted distributions.

1.2 Basic preliminaries and fundamental concepts

In order to discuss different properties associated with weighted distributions, it isfelt imperative to give the brief introduction of some preliminaries and fundamentalconcepts.

1.2.1 Some continuous distributions

This section consists of basic definitions of some continuous distributions that havebeen used in the succeeding chapters. These distributions include gamma, exponential,Maxwell, Pareto and power function distribution.

1.2.1.1 Gamma distribution

A random variable X is said to follow the gamma distribution with scale β and shapeα , i.e., X ∼ gamma(α,β ), if its probability density function (p.d.f.) is given by

f (x) =xα−1

β αΓ(α)exp(−x/β ), x≥ 0, α, β > 0. (1.1)

Gamma distribution is a special case of generalized gamma, whose p.d.f. is given by

fgg(x;α,β ,γ,µ) =

γ exp−(

x−µ

β

)γ(x−µ

β

)αγ−1

βΓ(α), x > µ ≥ 0,α,β ,γ > 0,

(1.2)where α, γ are two shape parameters, µ is location and β is the scale parameter.Density of two parameter gamma distribution can easily be obtained from (1.2) by

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1.2. Basic preliminaries and fundamental concepts

Gamma (α, β) Exponential (β) Chi Square (2α )

Moyal (µ, σ)

Erlang (α, λ)

Log Gamma (α, β, µ)

Gen. Extreme value (α, θ, µ=0) Inverse Gamma (α, β-1

)

Beta (a, b)

Pareto-II (λ=β1/ β2, 1, 0)

Pearson (type, a1, a0, b2, b1, b0)

Generalize Gamma (α, β, γ, µ)

Maxwell (θ) Rayleigh (θ)

Chi (υ) Nakagami (α, γ) Weibull ( β, γ, µ) Half Normal (θ)

β = 2 α = 1

γ =

1, µ

= 0

FIGURE 1.1: Relation of gamma distribution with other distributions.

substituting µ = 0 and γ = 1. Relation of gamma distribution with other distributionsis given in Figure 1.1.

1.2.1.2 Exponential distribution

A random variable X is said to follow one parameter exponential distribution with rateλ denoted by X ∼ exp(λ ), if its p.d.f. is given by

f (x; λ ) = λ exp(−λx), x≥ 0,λ > 0. (1.3)

The one parameter exponential distribution can be enlarged to the two parameterexponential distribution through the linear transformation Y = α +λX . The p.d.f. ofY is given by

fY (x; λ ,α) =

λ exp−λ (x−α), x≥ α

0, x < α,(1.4)

where α is known as the location parameter. If α = 0 and λ = 1, the distributionreduces to standard exponential distribution. The p.d.f. of an exponential distributionis monotonically decreasing and is having the thin tails, i.e., it decreases exponentiallyfor large values of x. Exponential distribution possesses one of the important property

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Gamma (α, λ)

Logestic (µ, β)

Weibull (α, β )

Extreme Value (α, β)

Erlang (α, λ)

Laplace ( µ, β)

Power Function (k, λ)

Rayleigh (σ )

Pareto (k, λ) Pearson (type, a1, a0, b2, b1, b0)

Exponential (λ)

Gumbel (α, β)

Frechet (α, β) Beta (λ, α)

𝒀 = 𝜶− 𝜷𝒍𝒐𝒈 𝒆−𝑿

𝟏− 𝒆−𝑿 ,𝝀 = 𝟏 𝑿 =

𝒀𝟐

𝟐, 𝝈 = 𝝀−𝟏

FIGURE 1.2: Relation of exponential distribution with other distribu-tions.

known as memoryless property due to which it is having the constant hazard rate.Relationship of exponential distribution with other distributions is given in Figure 1.2.

1.2.1.3 Maxwell distribution

A random variable X is said to have the Maxwell distribution, i.e., X ∼Maxwell(θ),if its p.d.f. is of the form

f (x; θ) =

√2π

θ3/2x2 exp

(−θx2

2

),x≥ 0,θ > 0. (1.5)

Maxwell distribution is a continuous probability distribution defined over the inter-val (0,∞) and parameterized by a positive real number θ known as rate parameter.The p.d.f. of Maxwell distribution is unimodal, possess thin tails and decreases expo-nentially for large values of x. Maxwell distribution can be derived form generalized

gamma distribution by substituting α =32,β =

√2θ ,γ = 2,µ = 0 in (1.2). Maxwell

distributions is also related to Chi-square distribution, i.e., if X ∼ Maxwell(1), thenX2 ∼ χ2

(3).

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1.2.1.4 Power function distribution

A random variable X is said to follow power function distribution with scale parameterθ and a shape parameter α denoted by X ∼ power(θ ,α) if its p.d.f. is given by

f (x;α,θ) =αxα−1

θ α, x ∈ (0,θ ],θ > 0,α > 0. (1.6)

The density of power function is monotone increasing in nature with global maximumoccurring at x = θ . Power distribution is closed under scaling, i.e., if X ∼ power(θ ,α),then for some p > 0, pX ∼ power(θ/p,α). It is also closed under maximum, i.e., ifX ∼ power(θ ,α) and X1∼ power(θ ,α1), then max(X ,X1)∼ power(θ ,α+α1). Powerfunction distribution with α = 1 reduces to U(0,θ) distribution. power(1,α) is aspecial case of Kumaraswamy distribution whose density is given by αβxα−1(1−xα)β−1,0< x< 1. If Y ∼ exp(α), then the transformation X =(θeY )−1∼ power(θ ,α).Inverse of power function random variable follows Pareto distribution.

1.2.1.5 Pareto distribution

Pareto distribution which is famous for its applications especially in economics wasintroduced by the Italian economist Vilfredo Pareto (1897), after discovering the be-havior of an income distribution and used it for modeling the income of Switzerland.A random variable X is said to follow Pareto distribution with scale parameter θ and ashape parameter k, denoted by X ∼ Pareto(θ ,k), if its p.d.f. is given by

f (x) =kθ k

xk+1 , x≥ θ ≥ 0, k > 0. (1.7)

P.d.f. of Pareto distribution decreases as a power law rather than exponential for largervalues of x and hence possess the fat tails. Pareto distribution is a continuous ana-logue of zeta distribution. If X follows Pareto(θ ,k), then X−1 and log(X/θ) followsrespectively power function and exponential distribution.

Different properties associated with the above discussed continuous distributionsare summarized in Table 1.3.

1.2.2 Approaches used for introducing a new distribution

During the last three decades or so, various approaches have been proposed to intro-duce new extensions of an already existing cumulative distribution G(x). Most of theseapproaches are given in the following general form.

Gnew(x) = B[G(x)], (1.8)

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where B(.) : [0,1]→ [0,1] is a bijective and monotonic function, Gnew(x) is the newextension of a parent/baseline distribution G(x). Therefore, from (1.8) one can gen-erate a vast family of distributions by choosing different baseline distributions withdifferent forms of B(.). There are various approaches of kind (1.8) and the first oneproposed during last three decades was due to Marshell and Olkin (1997). Marshelland Olkin (1997), defined the function B(.) in the following way in order to add onemore parameter α to the baseline distribution G(x).

B[G(x)] =αG(x)

1− (1−α)G(x),α > 0. (1.9)

The new distribution generated by using (1.9) is called Marshall-Olkin-G distribution.Many approaches using on (1.8) have been proposed and are worth to mention. Theseapproaches are exponentiated-G distributions due to Gupta, Gupta, and Gupta (1998),beta-G distributions (Eugene, Lee, and Famoye (2002)), gamma-G distributions (Zo-grafos and Balakrishnan (2009)), Kumaraswamy-G distributions (Cordeiro and Castro(2011)), generalized beta-G distributions (Alexander et al. (2012)), beta-extended-Gdistributions (Cordeiro, Silva, and Ortega (2012)), gamma-exponentiated-exponentialdistributions (Ristic and Balakrishnan (2012a)), gamma uniform-G distribution (Torabiand Montazeri (2012)), T-X family of continuous distributions (Alzaatreh, Lee, andFamoye (2013)), exponentiated generalized class of distributions (Cordeiro, Ortega,and Cunha (2013)), exponentiated-Kumaraswamy-G distributions (Lemonte, Barreto-Souza, and Cordeiro (2013)), geometric exponential-Poisson-G distributions (Nadara-jah, Cancho, and Ortega (2013)), log gamma generated family of distributions (Amini,Mostafaee, and Ahmadi (2014)), truncated-exponential skew-symmetric-G distribu-tions (Nadarajah, Nassiri, and Mohammadpour (2014)), modified beta-G distributions(Nadarajah, Teimouri, and Shih (2014)), and exponentiated exponential-Poisson-G dis-tributions (Ristic and Nadarajah (2014)) etc. In order to see the structure of differentB(.) functions which are used to introduce the above mentioned G distributions, onecan go through Nadarajah and Rocha (2016). Nadarajah and Rocha (2016) suggestedan R-package titled "Newdistns" in which programming for different statistical mea-sures associated with nineteen newly developed extensions were discussed in detail,may it be programming for computing the probability density, cumulative distribution,quantile, generation of random numbers or estimation of parameters.

1.2.2.1 Transmuted family of distributions

Shaw and Buckley (2009) suggested a technique known as transmutation, the noveltyof which is that it can be used to introduce skewness or kurtosis into a symmetric or

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other type of distribution. Shaw and Buckley (2009) introduced two types of compositemaps, viz., sample transmutation maps (STMs) and rank transmutation maps (RTMs)after considering two cumulative distribution functions F and G. These transmutationmaps are the functional composition of cumulative distribution function of a randomvariable with the quantile function of another random variable. The general structureof these two maps are y = G−1[F(x)] (STMs) and z = G[F−1(x)] (RTMs). These mapsproved to be very interesting in discovering vast family of Non-Gaussian distributionsand hence made statisticians to pay their attention towards the importance of thesemaps in statistical literature, see, e.g., Gilchrist (2000), Aryal and Tsokos (2011), Aryal(2013), Khan and King (2013), Lucena, Silva, and Cordeiro (2015), Elbatal and Aryal(2016), Haq et al. (2016), Merovci et al. (2017) and Deka, Das, and Baruah (2017) etc.Some of the parametric distributions that were generated in Shaw and Buckley (2009)are skew-uniform, skew-exponential, skew-normal, and skew-kurtotic-normal. STMsand RTMs can be used to modify the moments associated with skewness and kurtosisof an underlined distribution.

To obtain the density function of Y after changing a variable X to Y through thetransformation Y = g(X), the most reliable procedure is to obtain it via distributionfunction rather than density. Keeping these things in view, Shaw and Buckley (2009)attempted to infer the corresponding change in variables X and Y which are linkedthrough Y = g(X) and are having distribution functions FX(x) and GY (y) respectively.Shaw and Buckley (2009) tried to invent prosaic changes of variable, that can be ap-plied to the ranks of random variable, so as to produce a modulation by introducing theskewness and kurtosis in a distribution. Two types of transmutations maps discussedby Shaw and Buckley (2009) are given as follows.

1.2.2.1.1 Sample transmutation mapping. The sample transmutation mapping (STM) between two distribution functions say FX(x) and GY (y) is give by

Ts(G−1Y (U)) = F−1

X [U ], i.e., Ts(y) = F−1X [GY (y)] = QF(GY (y)), 0≤U ≤ 1, (1.10)

where QF(.) is the quantile function corresponding to FX(x). The sample transmutationmapping given in (1.10) transmutes quantile from GY (y) into the quantile from FX(x).

1.2.2.1.2 Rank transmutation mapping. Let F1(x) and F2(x) be two different dis-tribution functions with same support set, then rank transmutation mapping (RTM) isdefined as

GR12(u) = F2(F−11 (u)), GR21 = F1(F−1

2 (u)). (1.11)

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RTM maps the unit interval I = [0,1] into itself and satisfies Gi j(0) = 0, Gi j(1) = 1.

1.2.2.1.3 Quadratic rank transmutation mapping. The simplest form of ranktransmutation mapping is Quadratic rank transmutation mapping (QRTM) which isdefined as

Gt(x) = (1+β )G(x)−βG(x)2, −1≤ β ≤ 1, (1.12)

where Gt(x) is known as the transmuted version of baseline distribution function G(x).The corresponding density function is given by

gt(x) = (1+β )g(x)−2βG(x)g(x), −1≤ β ≤ 1. (1.13)

1.2.2.2 Transformed-Transformer (T-X) family of distributions

To generate the beta-generated family of distribution due to Eugene, Lee, and Famoye(2002) and Kumaraswamy G family of distribution due to Cordeiro and Castro (2011)one is confined to use the generator with support [0,1]. This confinement raised manyinteresting questions in the minds of Alzaatreh, Lee, and Famoye (2013) especiallyif one can use generator with support other than [0,1]. Alzaatreh, Lee, and Famoye(2013) addressed the same question and suggested a new technique to derive a familyof distributions by using any density function as a generator. The family of distributionderived by using this technique is termed as the Transformed-Transformer family orsimply T -X family.

Let F(x) be the cumulative distribution function (c.d.f.) of a continuous random X

and v(t) be the p.d.f. of another random variable T with support T ∈ [a,b], −∞≤ a <

b≤ ∞ . If ∃ a function say W (.) satisfying the following conditions.

(a) W (F(x)) ∈ [a,b].

(b) W (F(x)) is differentiable and monotonically non-decreasing function.

(c) W (F(x))→ a as x→−∞ and W (F(x))→ b as x→ ∞.

Then, according to Alzaatreh, Lee, and Famoye (2013) the distribution and densityfunction of T -X family are respectively given by (1.14) and (1.15).

G(x) =∫ W (F(x))

av(t)dt = Pr[T ≤W (F(x))] =V (W (F(x))). (1.14)

g(x) =

ddx

W (F(x))

vW (F(x)) , (1.15)

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TABLE 1.1: W (.) functions suggested by Alzaatreh, Lee, and Famoye(2013) for the support T ∈ [a,∞) and T ∈ (−∞,∞)

Support of Transformed W (F(x)) g(x)

[0,∞)F(x)

1−F(x)f (x)

[1−F(x)]2v(

F(x)1−F(x)

)

[0,∞) − log[1−Fα(x)]α f (x)Fα−1(x)

1−Fα(x)v(− log(1−Fα(x)))

[0,∞)Fα(x)

1−Fα(x)α f (x)Fα−1(x)[1−Fα(x)]2

v(

Fα(x)1−Fα(x)

)

(−∞,∞) log[− log(1−F(x))]f (x)v(log(− log(1−F(x))))(F(x)−1) log(1−F(x))

(−∞,∞) log[

F(x)1−F(x)

]f (x)

F(x)(1−F(x))v(

log(

F(x)1−F(x)

))

(−∞,∞) log[− log(1−Fα(x))]α f (x)Fα−1(x) v(log(− log(1−Fα(x))))

(Fα(x)−1) log(1−Fα(x))

(−∞,∞) log[

Fα(x)1−Fα(x)

]α f (x)

F(x)(1−Fα(x))v(

log(

Fα(x)1−Fα(x)

))

where V (t) = V (T ≤ t) is the c.d.f. of T . From (1.14), it can be seen that thedistribution function of T -X family is a composition of V,W and F , i.e., G(x) =

(V o(Wo(F)))(x).The family of distribution given in (1.14) is termed as transformed-transformer

because of the fact that random variable X is taken as transformer to transform an-other random variable T known as transformed through the function W (.). The mostinteresting property of T -X family is that it can be used to generate a vast variety ofprobability distributions on considering different combinations T, X and W (.). Forexample if T ∈ [0,1] and W (F(x)) = F(x) or W (x) = [F(x)]α , then the correspond-ing T -X family will be same as the beta- generated family of distribution which wassuggested by Eugene, Lee, and Famoye (2002). Moreover, if W (F(x)) = F(x) andT ∈ [a,b], then the density given in (1.15) reduces to

g(x) =

ddx

F(x)

vF(x)= f (x)v(F(x)), (1.16)

which is same as the density of univariate skew distribution family introduced by Fer-reira and Steel (2006) with weight function v(.).

Some of the W (.) functions suggested by Alzaatreh, Lee, and Famoye (2013) forthe support sets T ∈ [a,∞) and T ∈ (−∞,∞) are given in Table 1.1

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1.2.3 Basic reliability concepts

Study of a lifetime and its distribution is termed as reliability or survival analysis. LetT be a nonnegative and absolutely continuous random variable denoting the lifetimeof an item. Then, the various statistical properties associated with T can be understoodby making the use of following underlined functions.

1.2.3.1 Reliability function

Reliability function also known as survival function denoted by R(t) represents theprobability that an item survives or works at least up to a time point t.

R(t) = Pr(T ≥ t) = 1−Pr(T ≤ t) = 1−F(t), (1.17)

where F(t) is the distribution function of T .

1.2.3.2 Probability density function

The probability density function (p.d.f.) of a lifetime T represents the ratio of proba-bility that an item fails in an infinitesimal interval (t,∆t) to the length of interval (∆t),i.e.,

f (t) = lim∆t→0

Pr(t < T ≤ ∆t)∆t

. (1.18)

1.2.3.3 Hazard rate

The hazard rate also known as failure rate, denoted h(t) is defined as the probability offailure of an item during an infinitesimal time interval (t, ∆t), given that the individualhas survived to the left extreme of the interval (i.e., t).

h(t) = lim∆t→0

Pr(t < T ≤ t +∆t|T > t)∆t

= lim∆t→0

Pr(t < T ≤ t +∆t)∆t Pr(T > t)

=f (t)R(t)

. (1.19)

1.2.3.4 Cumulative hazard rate

Cumulative hazard rate of T is given by

H(t) =∫ t

0h(x)dx. (1.20)

1.2.3.5 Mean residual life

Mean residual life of an item having age t is defined as the expected value of theremaining life after time t. Mean residual life (m.r.l.) at time T = t is denoted by m(t)

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and is given by

m(t) = E[T − t|T > t] =∫

∞

t R(x)dxR(t)

. (1.21)

Both m.r.l. and hazard rate function are conditional concepts conditioned on survivalto a time point t. The difference between two is that hazard rate is instantaneous andprovides information about an infinitesimal interval just after t whereas m.r.l. gives in-formation about the remaining life after t. For more details one can go through Barlowand Proschan (1965). If m.r.l. and hazard rate both exist then following relationshipholds between them, see Guess and Proschan (1988).

m′(t) = m(t)h(t)−1. (1.22)

1.2.4 Order statistics

The literature related to order statistics is so vast that it finds its existence in astron-omy when astronomers were interested in the estimation of location beyond ordinarydescriptive statistics like mean. The formal investigation of order statistics dates backto the work of Tippett (1925) when he derived the c.d.f. of largest order statistics fromstandard normal distribution and found the mean of sample range. However, Wilks(1942) used term order statistics for the first time. There is a long history of orderstatistics, one may refer David (2006), Arnold, Balakrishnan, and Nagaraja (1992).

Let X1,X2, ...,Xn be a random sample of size n based on a continuous random vari-able X with p.d.f. f (x) and c.d.f. F(x). If this sample is arranged from smallest tolargest observation as X1:n ≤ X2:n ≤ ...≤ Xn:n, then Xr:n, r = 1,2, ...,n is called the rth

order statistics which is the rth smallest value in a randomly selected sample of size n.The value of Xr:n changes randomly with the change in sample and it’s density functionis given by

fr:n(x) =n!

(r−1)!(n− r)![F(x)]r−1 f (x)[1−F(x)]n−r. (1.23)

The density of smallest (X1:n) and largest (Xn:n) order statistics is obtained from (1.23)simply by putting r = 1 and r = n as

f1:n(x) = n f (x)[1−F(x)]n−1. (1.24)

fn:n(x) = n[F(x)]n−1 f (x). (1.25)

The densities given by (1.24) and (1.25) can also be derived from the density of expo-nentiated version of X which was suggested by Cordeiro, Ortega, and Cunha (2013)and is given by

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fexp(x) = αβ [1−F(x)]α−1[1−1−F(x)α ]β−1 f (x), −∞ < x < ∞, α > 0, β > 0.(1.26)

It is clear that the p.d.f. of X1:n and Xn:n can be obtained from (1.26) by substitutingα = n,β = 1 and α = 1,β = n respectively.Joint density of Xr:n and Xs:n (i.e., rth and sth order statistic) with r ≤ s is given by

fs,r:n(xr,xs) =n!

(r−1)!(s− r−1)!(n− s)![F(xr)]

r−1[F(xs)−F(xr)]s−r−1

× [1−F(xs)]n−s f (xr) f (xs),−∞ < xr < xs < ∞. (1.27)

If the size of sample is odd, i.e., n = 2m+1,m = 0,1, ..., then (m+1)th order statisticsis the sample median (X) and its density is given by

fx(x) =(2m+1)!

m!m![F(x)]m f (x)[1−F(x)]m. (1.28)

However, if sample size is even, i.e., n = 2m, m = 0,1,2..., then the sample median(X) is given by X = (Xm:n +Xm+1:n)/2 and its density is given by

fx(x) =2m!

(m−1)!(m−1)!

∫ x

−∞

[F(xm)]m−1[1−F(2x− xm)]

m−1 f (xm) f (2x− xm)dxm,

−∞ < xm < x < xm+1 < ∞. (1.29)

Order statistics plays a vital role in statistical analysis especially in nonparametricproblems Reiss (2012). It finds its application in extreme value theory and is used toestimate the location and scale parameter. Applications of order statistics in a broadersense can be understood from robust location estimation, detection of outliers, cen-sored sampling, reliability, quality control, procedure of selecting the best, inequalitymeasurement, Olympic records, allocation of prize money in tournaments see Arnold,Balakrishnan, and Nagaraja (1992). Considerable research has been devoted to un-derstand the applications of order statistics by different researchers (Balakrishnan andRao (1998)).

1.2.5 Measures of inequality

Measure of inequality describes the extent to which a variable under study is distributedunevenly among different proportions of statistical population. If every proportion ofthe population shares or receives same proportion of an underlined variable then sucha case is known as the case of perfect equality. Similarly, perfect inequality is defined

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as a situation wherein a single unit of the population possesses/receives entire amountof the variable under consideration. Although there had been attempts to develop amathematical tool for measuring the extent of inequality, the first major success wasLorenz curve due to American economist Lorenz (1905), in the field of economics.Since 1905, Lorenz curve is widely used to study the distribution of income. It isimperative to study the inequality because of the fact that disparity in living standardof population is the outcome of income inequality. Therefore, one can reduce thedisparity in a population by reducing the inequality. The concept of inequality doesn’tremain confined to the study of disparity in income only but is now analogously usedin different areas, e.g., Damgaard and Weiner (2000), Dixon et al. (1987), Andel et al.(1984), Sun et al. (2010), Vasa, Lumpe, and Branch (2009),Brown (1994), Groves-Kirkby, Denman, and Phillips (2009), Groot (2010) etc. There exist various tools instatistical literature that are used to study the disparity, we discuss some of the wellknown and widely used measures of inequality:

1.2.5.1 Lorenz curve and Bonferroni curve

In economics, Lorenz curve is a graphical representation of the size distribution of in-come or wealth. It was an American economist Max O. Lorenz (1905), who developedLorenz curve as a tool for making comparison in income distributions of a populationat different time points, or between different populations at the same epoch of time.The Lorenz curve is usually represented by a function L(p), where p is the cumulativeproportion of income units, represented by the horizontal axis. L(p) denotes the cumu-lative proportion of the total wealth or income received by units, arranged from low tohigh income and represented by the vertical axis. Lorenz curve and the associated Giniindex are off course the most popular indices of income inequality. However, there aresome measures which despite possessing interesting characteristics are not used oftenfor measuring income inequality. Bonferroni curve is one such measure, which havethe advantage of being represented graphically in the unit square and can also be re-lated to the Lorenz curve see Giorgi and Mondani (1995). The Bonferroni and Lorenzcurves are not only used in economics in order to study the relation between incomeand poverty, it is also being used in reliability, medicine, insurance and demography.The Bonferroni and the Lorenz curves for a non-negative random variable X with den-sity function f (x) and distribution function F(x) are respectively given by (1.30) and(1.31).

B(p) =1

pµ

∫ q=F−1(p)

0x f (x)dx. (1.30)

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0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

p

L(p)

Egalitarian Line / Line of perfect equalityPopulation−IPopulation−IIPopulation−III

FIGURE 1.3: Lorenz curve

L(p) = pB(p) =1µ

∫ q=F−1(p)

0x f (x)dx, (1.31)

where q = F−1(p) and µ = E [x]. From (1.31), it is obvious that L(p) = 0 when p = 0and L(p)= 1 when p= 1. If f (x) is continuous, then by using the fundamental theoremof calculus the first and second order derivative of L(p) w.r.t. p are given as follows.

dL(p)d p

=1µ

[F−1(p) f (F−1(p))

] dF−1(p)d p

=1µ

[F−1(p) f (F−1(p))

f (F−1(p))

]=

qµ. (1.32)

d2L(p)d p2 =

1µ

dF−1(p)d p

=1µ

1f (F−1(p))

=1

µ f (q). (1.33)

From (1.32) and (1.33), it can be seen that both the slope and second order derivativeof L(p) are positive for all the possible values of p. Therefore, it can be concluded thatLorenz curve is monotonically increasing function of p and is convex to the horizontalaxis, implying L(p)≤ p. Line for which L(p) = p is called egalitarian line or line ofperfect equality, otherwise it is known as line of inequality or Lorenz curve.

1.2.5.2 Gini index

Gini-index, one of the widely used measure of inequality is attributed to Corrado Gini(1912) and is defined as the ratio of the area between Lorenz curve and egalitarian lineto the area between egalitarian line and line of perfect inequality. It is also known asknown as Gini-coefficient. More the value of Gini-coefficient more is the extent ofinequality. There exist different formulation of Gini-index which are mostly unknown

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today that is the reason Ceriani and Verme (2012) summarized thirteen different formu-lations as a mark to celebrate the 100th anniversary of Gini-index. For further detailson various measures of inequality one may refer to Kakawani (1980) and Atkinson andBourguignon (2014).

1.2.6 Entropy

One of the key component in information theory is the concept of entropy. Informationtheory deals with the study of transmission, processing, utilization, and extraction ofinformation. It was Shannon (1948), who originally proposed this theory in a land-mark article, after viewing information as the resolution of uncertainty. Information isthought of as a set of possible messages where the goal is to send these messages over anoisy channel and then to have a receiver reconstruct the message with low probabilityof error in spite of the channel noise. The quantification, storage and communicationof information is having a key measure known as entropy. The amount of uncertaintyin the value of a random variable or the outcome of a random process is measured interms of entropy measure. There exist various entropy measures in literature but themost popular and widely used are given as.

1.2.6.1 Renyi entropy

Information measure proposed by Alferd Renyi (1961) known as Renyi entropy oforder δ associated with a random variable X ∼ f (x) is given by

HR (δ ) =1

1−δlog[∫ f (x)δ dx

], δ ≥ 0,δ 6= 1. (1.34)

1.2.6.2 Shannon entropy

Shannon’s measure of information is given by

HS = E [− log f (x)] . (1.35)

Shannon entropy is the limiting case of Renyi entropy, i.e.,

HS = limδ−→1

HR (δ ) . (1.36)

1.2.7 Simulation techniques

To support the application and validity of a newly introduced model, situations need tobe traced under which its adequacy and importance is justified. It may or may not be

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possible to trace such an ideal situations due to the reason that there are enormouslya huge number of random phenomena in the universe giving birth to different typesof data sets and problems. Thus instead of tracing a particular situation, an environ-ment favorable for the execution and implementation of newly introduced model ortechnique is created, so that its validity and adequacy is shown. The creation of sucha favorable environment is known as simulation. Thus, in order to show the applica-bility of a newly introduced probability model one needs to generate/simulate randomnumbers from it. Though there exist a number of techniques one can use to simulaterandom numbers from an underlined distribution, we have used the method of inversesampling and rejection method in the subsequent chapters.

1.2.7.1 Inverse sampling method

In inverse sampling method (inverse transformation method) a random number from adistribution function F(x;Θ) is generated by solving the equation F(x;Θ) = p for x, atpreassigned values of Θ and p, where p ∼ U(0,1). Distribution function being bijec-tive in nature with quantile function (F−1(p)) as its inverse, thus for each value of p

and at a preassigned value of Θ, ∃ a unique solution which is given by x = F−1(p, Θ).Therefore, to generated a random sample of size n from a particular distribution atΘ = Θ

′(say), F(x, Θ

′) = p is solved for x at n independent values of p. The solutions

obtained at p = 0, p = 0.25, p = 0.5, p = 0.75 and p = 1, are respectively minimum,first quartile (Q1), median (Q2), third quartile (Q3) and maximum. This method isgenerally used to generate random numbers from almost all kinds of continuous distri-butions. Preposition which provides the basis for this method can be given as:Proposition 1.1 Let p ∼ U(0,1). Define a random variable X = F−1(p, Θ) where,F−1(p, Θ) be a quantile function with Θ as its parameter space. Then the randomvariable X has the distribution function F(X ,Θ).Proof: Let us suppose that FX be the distribution function of random variable X . There-fore, for X = a we can write

FX(a) = Pr(X ≤ a) = Pr(F−1(p, Θ)≤ a).

Since, F−1(p, Θ)≤ a ⇐⇒ p≤ F(a, Θ) because F(a, Θ) is monotone. Therefore,

FX(a) = Pr(p≤ F(a, Θ)) = F(a, Θ).

1.2.7.2 Rejection method

Rejection method which forms the basis for Metropolis like algorithm, is a technique ofgenerating random numbers from a target distribution Ft(x,Θ) seeking the utilization

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of another distribution Fp(x,Θ1) known as proposal or instrumental distribution. It isalso known as Von-Nueman sampling after the name of Hungarian-American math-ematician John von Neumann. The sampling is based on a type of Monte Carlomethod known as accept-reject algorithm that is why it is also called acceptance-rejection method. Rejection method works for any kind of distribution in Rn witha density ft(x,Θ). The observation (Y = y) simulated from the proposal distributionis accepted as a an observation from the target distribution with a probability propor-tional to ft(x; Θ)/ fp(x; Θ1), where ft(x; Θ) and fp(x; Θ1) are respectively the targetand proposal density.Specifically, let m be a constant such that

ρ(y) =ft(y)fp(y)

≤ m ∀ y. (1.37)

where, m is the maximum of ratio of two densities determined by solving the followingequation.

∂ρ(y)∂y

= 0 s.t.→

∂ 2ρ(y)∂y2 |y=m< 0. (1.38)

After computing the value of m, rejection method consists of followings two steps tobe followed.

i) Simulate random numbers Y and U respectively from the proposal density andU(0,1) density.

ii) Retain Y and set X = Y if U ≤ ρ(y)m

otherwise return to step (i).

Proposition 1.2 The random variable Y generated from proposal density fp(y; Θ1)

and retained under rejection method has density function ft(x; Θ).Proof: Let us suppose Y be the value generated from proposal density therefore,

Pr(Y ≤ y) = Pr(X ≤ y|U ≤ ft(X)/m fp(X))

Pr(Y ≤ y) =Pr(X ≤ y,U ≤ ft(X)/m fp(X))

Pr(U ≤ ft(X)/m fp(X))

Pr(Y ≤ y) =∫

Pr(X ≤ y,U ≤ ft(X)/m fp(X)|X = x) fp(x)dxK

Pr(Y ≤ y) =∫ y−∞( ft(x)/m fp(x)) fp(x)dx

K

Pr(Y ≤ y) =∫ y−∞( ft(x)/m)dx

K

Pr(Y ≤ y) =∫ y−∞ ft(x)dx

mK.

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As limy→∞, we get K = 1/m. Hence, Pr(Y ≤ y) =∫ y−∞ ft(x)dx.

1.2.8 Integration using Monte Carlo simulation

Sometimes it becomes very hectic to compute the numerical value of a definite integraland therefore a technique known as Monte Carlo integration is used for its numericalcomputation. This integration technique is used to compute definite integral throughthe process of random number generation from a density, the support of which needsto be same as the range of integral. Consider the following general form of a definiteintegral. ∫ b

am(x)dx. (1.39)

In Monte Carlo integration the numerical approximation of (1.39) is found bychoosing a density function say f (x) whose support is [a,b]. After choosing f (x),(1.39) is rewritten as follows.

∫ b

am(x)dx =

∫ b

a

m(x)f (x)

f (x)dx =∫ b

a¯h(x) f (x)dx = E f (x)[ ¯h(x)]≈

∑ni=1 ¯h(xi)

n, (1.40)

where xi, i= 1,2, ...,n is a random sample of size n which is generated from the densityfunction f (x). Thus, as a consequence of strong law of large numbers, larger thesample size more approximate value of the definite integral one can have.

1.2.9 Partial ordering

Let f (x) and g(y) be the densities of two random variables X and Y respectively. Letthe corresponding distribution functions be F(x) and G(y). hX(x) and hY (y) be therespective hazard rate functions and the corresponding mean residual life function bemX(x) and mY (y). Then one can define the following order relations between X and Y

see Lai and Xie (2006), Shaked and Shanthikumar (2007) and Jain, Singla, and Gupta(2014).

i) Stochastic ordering: X is said to be smaller than Y stochastically denoted byX ≤ST Y if F(x)≤ G(x)∀x≥ 0.

ii) Failure rate ordering: X is said to be smaller than Y in terms of failure rate and

is denoted by X ≤FR Y ifF(x)G(x)

is decreasing in x, where F(x) and G(y) are the

survival functions of X and Y .

iii) Likelihood ratio ordering: X is said to be smaller than Y in likelihood ratio

order, i.e., X ≤LR Y iff (x)g(x)

is decreasing in x.

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iv) Mean residual life ordering: X is said to be smaller than Y in terms of mean

residual life, i.e., X ≤MR Y if∫

∞

x F(x)∫∞

x G(x)is decreasing in x.

The implication between above defined partial orderings are given by the followingchain.

X ≤LR Y =⇒ X ≤FR Y =⇒ X ≤ST Y

(1.41)

X ≤MR Y

1.2.10 Estimation

There are different methods which are used to estimate a population parameter either inthe form of single value or in the form of an interval and according theory of estimationis classified into two categories, i.e., point and interval estimation.

1.2.10.1 Point estimation

Estimation procedure used to find the estimate of a parameter in the form of a singlevalue is called point estimation. There are various methods which are employed tofind the point estimator. The most popular and widely used methods are maximumlikelihood estimation method and method of moments.

1.2.10.1.1 Method of Maximum likelihood estimation: It is apparent from thename that maximum likelihood estimation method yields an estimator at which like-lihood function attains its maximum. Let us suppose X be a random variable withdensity f (x;Θ), where Θ = (θ1,θ2, ...,θk) is a k-dimensional parameter vector. There-fore the M.L.E. of Θ usually denoted by Θmle = (θmle1 , θmle2 , ..., θmlek) is obtained bysolving the following system of equations.

∂ log[l(Θ | x)]∂θi

= 0 s.t.→

∂ 2 log[l(Θ | x)]∂θ 2

i< 0, i = 1,2, ...,k, (1.42)

where log[l(Θ | x)] = ∑ni=1 log[ f (xi;Θ)] and x = (x1,x2, ...,xn) denotes respectively the

log-likelihood function and a random sample of size n. In general, M.L.E. is not an un-biased estimator but it is better to call it almost unbiased and consistent. For instance,in case of normal law the M.L.E. of σ2 is σ2 = n−1

∑ni=1(xi−x)2 which is an unbiased

estimator of σ2 with a negative bias of n−1σ2 while the unbiased estimator of σ2 is(n−1)−1

∑ni=1(xi−x)2. It is obvious that on increasing n, the bias (−n−1σ2) will tend

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to zero. Thus, the use of term almost unbiased and consistent is justified. One of theimportant property of M.L.E. is invariance, i.e., if Θmle is the M.L.E. of Θ then theM.L.E. of any transformation g(Θ) is g(Θmle).

1.2.10.1.2 Method of moments: Let X1,X2, ...,Xn be a random sample from a pop-ulation with p.d.f. or p.m.f. f (x,Θ), where Θ = (θ1,θ2, ...,θk). Then, the momentestimator of Θ, i.e., Θmm = (θ1, θ2, ..., θk) is obtained by solving the following systemof k equation.

n

∑i=1

X ri

n=∫

∞

−∞

xr f (x)dx orn

∑i=1

X ri

n= ∑

xxr f (x), r = 1,2, ...,k, (1.43)

where ∑ni=1

X ri

nis the rth sample moment and

∫∞

−∞xr f (x)dx or ∑x xr f (x) is the rth pop-

ulation moment.

1.2.10.2 Interval estimation

The concept of interval estimation came into existence after J. Neyman (1937) triedto determine the bounds of an interval that would contain the true value of parametersay θ so that the greatest possible accuracy of results can be assured. The accuracyof an estimator T was described by deriving another estimator ST , which is standarddeviation of the sampling distribution of original estimator T . After having ST , theinterval estimate of θ is written in the form T ±mST , where m is any constant. Thus,the two bounds which were described and discussed in detail by J. Neyman (1937) areθ = T −mST and θ = T +mST . Smaller the value of ST more accurate will be theestimate that is why M.V.U.E. is one of the preferred estimator. An interval estimate iscalled by different terms, for instance, in frequentest terminology it is called confidenceinterval, in Bayesian setup it is known as credible interval and in regression analysis istermed as tolerance interval etc. However, there are various prevalent methods used toderive an interval estimator, we will discuss only the following method in brief.

1.2.10.2.1 Interval estimation using the asymptotic normal property of M.L.E.One of the well known property of M.L.E. (Θmle) of parameter Θ = (θ1,θ2, ...,θk)

is that, it is consistent and almost unbiased estimator under large sample size n, i.e.,on increasing n, Θmle converges in probability to its true value (Θ) and E[Θmle] = Θ.The difference between true value of Θ and Θmle converges in distribution to a normaldistribution, i.e.,

√n(Θmle−Θ)

d−→ N(O,I−1(Θmle)

), (1.44)

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where I−1(Θ) is the inverse of Fisher’s information matrix I(Θ) which is given by

I(Θ) =−E

∂ 2 log l (Θ|x)∂θ 2

1. . .

∂ 2 log l (Θ|x)∂θ1∂θi

...

. . .

...

∂ 2 log l (Θ|x)∂θi∂θ1

. . .∂ 2 log l (Θ|x)

∂θ 2i

, i = 1,2, ...k. (1.45)

Therefore 100(1−α)% confidence interval for Θ is given by

Θ ∈[

Θmle± zα/2

√diag

(I−1(Θmle)

)], (1.46)

i.e.,

θi ∈[

θmlei± zα/2

√I−1

i,i (θmlei)

], i = 1,2, ...,k, (1.47)

where I−1i, j represents the element belonging to ith row and jth column of I−1(Θmle).

1.2.11 Comparison criteria and goodness of fit

In every statistical modeling problem, one of the key question that needs to be an-swered is: which model among a class should be selected? so that the inference/predict-

ion can be drawn with much higher accuracy. Thus, the selection and specificationof a model is of a big concern before drawing a prediction and probability associatedwith that prediction. There exist various statistical tools and techniques which helpone to select a model of best fit among a class of models for modeling a particularrandom phenomena. Some of the tools that are used for model selection are discussedas follows:

1.2.11.1 Akaike information criterion

Akaike information criterion (AIC) is attributed to Japanese statistician H. Akaike(1974) and is given by

AIC =−2log[l(Θmle|x)]+2k, (1.48)

where k denotes the number of parameters to be estimated in a model and l(Θmle|x) isthe maximum likelihood. However, AIC is widely used as a tool for model selectionbut becomes inadequate to use under small sample size. In small size problems, AIC

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substantially identifies the model with too many parameters to be selected. Therefore,in case of small sample size it is preferred to use Akaike information criterion corrected(AICc) which is an extension of AIC with an amount of extra penalty and is given as

AICc = AIC+2k2 +2kn− k−1

. (1.49)

It is apparent from (1.49) that on increasing the sample size , AICc will converge toAIC as the penalty term involves n in the denominator. Among a class of models,model which possesses minimum AIC or AICc value is considered to be the modelof best fit. Thus, given a class of models, one should select the model with minimumvalue of AIC or AICc for modeling a particular data set.

1.2.11.2 Bayesian information criterion

Bayesian information criterion (BIC) was suggested by G. Schwarz (1978) and is de-fined as

BIC =−2log[l(Θmle|x)]+ k logn, (1.50)

where the notations carry their usual meaning as mentioned earlier. From a class of sayr models, the model which possesses minimum value of BIC is selected as the modelof best fit for some considered data set. For further details on the model selectioncriteria one may refer Burnham and Anderson (2007) and Ando (2010).

1.2.11.3 Kolmogorov Smirnov test

Kolmogorov Smirnov test which is also known as KS-test is a nonparametric test andwas introduced by A. N. Kolmogorov (1933) to test the goodness of fit for a singlesample. In 1939, N. V. Smirnov (1939) made it advanced by making it more applica-ble to test whether the difference/discrepancy between two empirical distributions orbetween an empirical and any reference distribution is significant or not.

Let Fe(x) and Fre f .(x) represent an empirical and a reference commutative distri-bution function respectively, then the KS-test statistic is given by

D = supx |Fe(x)−Fre f .(x)|, (1.51)

which is the supremum of absolute deviation between two distribution functions. Thereare two versions of KS-test, viz., one sample and two sample KS-test.

The test statistic given in (1.51) is used in case of one sample version of KS-testwhere the null hypothesis to be tested is "H0: Sample having an empirical distribution

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−2 −1 0 1 2 3 4

0.0

0.2

0.4

0.6

0.8

1.0

x

F(x

)=P

r.(X

≤x)

Fe1(x)

Fref.(x)

D = supx |Fe1(x) − Fe2

(x)|

(a) One sample KS-test

−1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

1.0

x

F(x

)=P

r.(X

≤x)

Fe1(x)

Fe2(x)

D = supx |Fe1(x) − Fe2

(x)|

(b) Two sample KS-test

FIGURE 1.4: Graphical representation of one sample and two sampleKS-test statistic D.

Fe(x) has came from the population with distribution function Fre f .(x)" against the al-ternative hypothesis "H1: it is not so". However, in case of two sample KS-test the nullhypothesis to be tested is defined as "H0: Two samples having empirical distributionsFe1(x) and Fe2(x) have came from the same population" against the alternative hypoth-esis "H1: they came form different populations" and accordingly the correspondingtest statistic is given by

D = supx |Fe1(x)−Fe2(x)|, (1.52)

Thus, KS-test is used to make comparison between two distribution functions andhelps to conclude whether the deviation between two distribution functions is signifi-cant or not. If the deviation D is significant then it is concluded that two distributionsare totally different at the desired level of significance. Syntax to be followed whileusing KS-test in R programming is given as

ks.test(x,y,alternative=c("two.sided","less","greater"))

ks.test(x,"cdf",pi,alternative=c("two.sided","less","greater"))

where "x" and "y" represent two numeric data sets whose empirical distributionfunctions are to be compared, "alternative" provides the option whether to usetwo sided or one sided version of KS-test, "cdf" is the R-name of reference distribu-tion function with "pi" as the value of its parameters, e.g., if reference distribution isN(2,5), then "cdf" will be replaced by "pnorm", "p1" by 2 and "p2" by 5. Afterperforming the test, value of test statistic D and the corresponding P-value is provided.

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Thus, larger the value of D implies large is the discrepancy between two distributionfunctions and consequently smaller will be the P-value. If P-value is less than 0.05then we reject the null hypothesis at 5% level of significance and conclude that the twodistribution functions are different.

1.3 Weighted distribution

The concept of weighted distributions can be traced from the work of Fisher (1934),wherein it is studied that how methods of ascertainment can influence the form ofdistribution of recorded observations. Later, it was developed and formulated in gen-eral terms by Rao (1965) in connection with modeling statistical data where the usualpractice of using standard distributions for the purpose was not found to be appro-priate. Thus, the concept of weighted distribution is attributed to Fisher (1934) andRao (1965) who first contemplated on the situations where observation fall in a nonexperimental, non-replicated and non-random categories. The best example of suchsituations is encounter sampling where an observation gets recorded only after one en-counters it. More generally one can say situations in which observations get recordedwith probabilities proportional to some weight function w(x). It is quite obvious thatwhile studying the real world random phenomena, the observations may be recordedwith an amount of inherent bias as a result of which these recorded observations willnot have the original distribution unless every observation is given actual chance toget recorded. Rao (1965) after studying similar situations laid down the concept ofweighted distributions to provide the unifying procedure for the problem of modelspecification and statistical inference.

Definition 1.1: Consider a random variable X which is generated by a naturalprocess and is supposed to have the density function f (x,θ),θ ∈ Ω, the parameterspace. To draw a random sample of n observations on X , one needs to use the samplingtechnique which will assign equal chance of getting selected to any observation on X

generated by the original process. Practically, it may happen that the observations areobserved with unequal probabilities instead of equal, e.g., encounter sampling. Let x

and y be two observations with w(x) : w(y) as their relative chance of being observed.Then the observed X denoted by Xω has the density which is given by

fw(x;θ) =w(x) f (x; θ)

E[w(x)], (1.53)

where w(x) ≥ 0 is a non-negative valued function and E[w(x)] =∫

w(x) f (x; θ) dx or= Σw(x) f (x; θ) depending on whether X is continuous or discrete random variable.

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1.3. Weighted distribution

TABLE 1.2: Some weight functions

x≥ 0 w(x) Referencesgeneral x Cox (1962)

continuous xω , ω > 0 Brown (1972), Patil and Ord (1976)eωx Patil and Ord (1976)

discrete

xω , ω > 0 Rao (1965), Patil and Ord (1976)x+1 Rao and Patil (1977)

x(x−1)...(x−ω +1) Patil and Ord (1976), Gupta (1975)ωx, 0 < ω < 1 Rao and Rubin (1964), Rao (1965), Kemp (1973)

1− (1−ω)x, 0 < ω < 1 Haldane, 1938, Rao (1965), Cook and Martin (1974)

The distribution given by (1.53) is known as weighted distribution with an arbitraryweight function w(x). However, if w(x) = xω then the resulting weighted distributionis termed as size-biased version of X or simply size-biased distribution. Moreover, ifw(x) = x and w(x) = x2, then the corresponding size-biased versions are respectivelyknown as length-biased and area-biased distribution. It is worth to mention that theidea of length biased sampling appeared first in Cox (1962).

The important component in the theory of weighted distributions is weight func-tion. There exist different forms of weight functions in the literature of weighted dis-tribution see for example Patil, Rao, and Ratnaparkhi (1986). These weight functionsare monotone in nature, either increasing or decreasing. Some of the weight functionused by different authors and mentioned in Rao and Patil (1977) are given in Table 1.2.

There is a lot of examples one can use to illustrate the circumstances responsiblefor generation of weighted distributions. Following are some of the examples whereapplication of weighted distributions can be ascertained.

i) Truncation: If a random variable X is truncated to a set A, then the applicationof weighted distribution is inevitable and the weight function need to be definedas:

w(x) =

1, x ∈ A

0, x /∈ A.(1.54)

A considerable research is devoted to the study of truncated distribution arisingout of non-ascertain ability of observation, see Patil (1959).

ii) Encounter sampling: There are situations in real life, wherein it is impossibleto adopt a scientific way of data collection rather an observation is recorded onlyif it gets encountered, e.g., in forestry an animal is recorded only if it is cited.Such a procedure of data collection is known as encounter sampling.

iii) Size biased sampling: Sampling where an observation gets recorded with prob-ability proportional to xω is known as size biased sampling of order ω .

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iv) Missing data: If a proportion 1− p(x), 0 ≤ p(x) ≤ 1, is missing out of naturalfrequency of x, then the suitable distribution used for analysis of recorded datais weighted distribution with weight function w(x) = p(x).

v) Partial destruction and damaged items: Situations exist in real life where anobservation is not ascertained fully or destructed under a destruction process.Such a situation is responsible for the alteration of original distribution and lead-ing to the generation of weighted distribution.

1.3.1 Connection between Bayesian inference and theory ofweighted distribution

The posterior density in Bayesian frame work can be connected to the concept ofweighted distributions by using the result of Mahfoud and Patil (1981). This resultshows that posterior density is the Bayesian analogue to the concept of weighted dis-tribution with likelihood playing the role of weight function and prior as parent dis-tribution. Consider X and θ as the underlined random variable and correspondingparameter respectively with f (x) and p(θ) being their respective densities. Therefore,in Bayesian frame work the posterior density is given by

π(θ |x) = f (x|θ)p(θ)f (x) =

∫f (x|θ)p(θ)dθ

=l(θ |x)p(θ)E[l(Θ|x)]

. (1.55)

Which is clearly a weighted distribution with likelihood as the weight function andprior as the parent density.

For further details on the concept of weighted distribution one may go through Rao(1965) wherein various situations responsible for the alteration of original distributionare discussed, like non-ascertain-ability of certain observations, partial destruction ofobservations and selection of observations with unequal probabilities. Rao and Patil(1977) and Patil (2002) illustrated situations responsible for generation of weighteddistributions. Various applications of weighted distributions were surveyed by Raoand Patil (1977), e.g., family size analysis, distribution of albinism, distribution of arare inherent trait in human heredity, study on alcoholism and family size, estimatingdensity of wildlife population, estimating the proportion of diseased plants, waitingtime paradox, renewal theory, cell cycle analysis and pulse labeling, efficiency of earlydetection for disease, forest production research, particle size distributions by thin sec-tion method, aerial survey in traffic research, rejection technique in random numbergeneration, weight size distribution in a breakage process and moment distribution ineconomics.

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1.4. Importance of the study

1.3.2 Some remarkable theorems related to weighted distributions

Let Xw denotes the weighted version of a random variable X under an arbitrary weightfunction w(x), then one can observe the following results.

Theorem 1.1 (Patil, Rao, and Ratnaparkhi (1986)). Xw ≥ ST X if w(x) is monotoneincreasing in x and Xw ≤ ST X if w(x) is monotone decreasing in x.

Theorem 1.2 (Patil, Rao, and Ratnaparkhi (1986)). E[Xw]> E[X ] if w(x) is monotoneincreasing in x and E[Xw]< E[X ] if w(x) is monotone decreasing in x.

Theorem 1.3 (Patil and Rao (1978)). Let the weight function w(x)> 0 and E[w(X)]<

∞, then

E[XW ]> E[X ] if cov[X ,w(X)]> 0 and E[XW ]< E[X ] if cov[X ,w(X)]< 0.

Theorem 1.4 (Patil and Rao (1978)). Let Xwi denotes the weighted version of a ran-dom variable X under the class of weight functions wi(x) > 0, i = 1,2,3, ... such thatE[wi(x)]< ∞, then for i 6= j = 1,2,3...

E[Xwi]> E[Xw j ] ifwi(x)w j(x)

is monotone increasing in x and

E[Xwi]< E[Xw j ] ifwi(x)w j(x)

is monotone decreasing in x.

Theorem 1.5 (Patil and Rao (1978)). Let E[X ]<∞ for a non-negative random variableX with Xw as its length biased version, then

E[Xw] = E[X ]+V[X ]

E[X ], where V(.) stands for variance and

[E[X−1w ]]−1 = E[X ], i.e., Harmonic mean of Xw is equal to mean of X .

Theorem 1.6 (Mahfoud and Patil (1982)). Under size biased sampling X is greaterthan its corresponding weighted version Xw in terms of failure rate, i.e.,

Xw ≤ FR X alsoV[X ]

E[X ]= E[Xw]− [E[X−1

w ]]−1.

1.4 Importance of the study

Sampling techniques play the fundamental role in carrying out any sort of statisticalinvestigation. Quite often, people don’t contemplate on the type of sampling technique

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used in extracting a part (sample) from the whole (population) due to which, most ofthe assumptions get violated and accordingly the results drawn may or may not bevalid anymore in real life. Thus, it is very imperative to look into the ways of samplingtechniques used in drawing a sample, before we go for statistical modeling. Situationsexist in real life where observations get selected with probabilities proportional to afunction, known as weight function, e.g., probability proportional to size, size biasedsampling and encounter sampling. It will not be genuine to carry out statistical mod-eling without taking into consideration the consequences of such situations.Thus, inorder to model data sets arising under such conditions, one must pay attention towardsthe concept of weighted distributions.

The concept of weighted distribution is very important because of the fact thatweighted distributions take into consideration the method of ascertainment by adjust-ing the probabilities of actual occurrence of events. We may arrive at wrong conclu-sions while failing to make such adjustment. Thus, it is very imperative to use theconcept of weighted distribution while dealing with a stochastic process in which theobservations are generated or recorded with varying probability. In order to increasethe accuracy and to draw sound results, our main motive becomes to give importanceto model specification. One of the unifying approaches for this purpose is to use theconcept of weighted distributions. The importance of weighted distributions can be un-derstood from Mcdonald (2010) discussing the need for teaching weighted distributiontheory.

There are some traditional theories and practices which have been occupied withreplication and randomization like environmentric theory. Observations also fall in thenon-experimental, non-replicated and non-randomized categories Patil (1991). Thus,our main interest lies in drawing the valid inference about random phenomena that ispossible only by making the use of suitable and flexible model for statistical modeling.Rao and Patil (1977) quoted "Although the situations that involve weighted distribu-tions seem to occur frequently in various fields, the underlying concept of weighteddistributions as a major stochastic concept does not seem to have been widely recog-nized". Thus, it is very essential to identify random processes where observations arerecorded with varying probabilities so that the validity and importance of weighted dis-tributions in statistical modeling can be understood. Weighted distributions has beenused in the selection of appropriate models for modeling the observed data, especiallywhen samples are drawn without a proper frame, see Patil and Taillie (1989).

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1.5. Review of literature

1.5 Review of literature

After the concept of weighted distribution came into existence, significant research hasbeen devoted to this area and the most recent work can be summarized as, for instance,Scheaffer (1972) generalized the concept of length biased sampling to size biased sam-pling. Cook and Martin (1974) estimated the visibility bias parameter and averagegroup size by adopting a model for quadrat sampling of randomly occurring groupof animals. Patil and Ord (1976) interrelated the concepts of weighted distributionsand size biased sampling, after identifying some situations under which the underlyingmodel is form-invariant. Patil and Rao (1978) studied weighted distributions and sizebiased sampling with applications to wildlife populations and human families. Simon(1980) described the problem of estimation and comparison of frequency of a charac-teristic in newly diagnosed patients based upon a length biased sample. Stene (1981)demonstrated how method of ascertainment generate different probability models fora sampled data set. Patil (1981) discussed the applications of weighted distributions instatistical ecology. Vardi (1982) studied nonparametric maximum likelihood estima-tion of life time distribution under length bias. In ecology, Dennis and Patil (1984) usedstochastic differential equations to arrive at weighted properties of size-biased gammadistribution. Rao (1985) studied the weighted distributions arising out of method ofascertainment. Drummer and McDonald (1987) derived the estimators of total popula-tion size using the theory of weighted distributions in line transect sampling. Janardanand Rao (1987) studied the Lagrange distributions and showed that for a particularform of the weight function and under some conditions, the weighted versions of La-grange distributions of the first kind belong to the class of Lagrange distributions of thesecond kind. Ramsey et al. (1988) gave a brief overview about transect sampling andillustrated the techniques using data related to terrestrial and marine animals. Patil andTaillie (1989) reviewed various properties of univariate and bivariate weighted distri-butions and observed the effect of weighted observations on Bayesian analysis. Patiland Taillie (1989) also discussed some applications of weighted distributions. Thefindings of Bergstedt and Anderson (1990) advocates the use of remote sensing andline transect sampling for obtaining the reliable estimates of object density on a lakebed. Jones (1991) proposed a kernel density estimator for length biased data. Arnoldand Nagaraja (1991) discussed some properties of bivariate weighted distributions.Power (1992), studied the consequences of sampling bias in context of distribution ofoil and gas pools. Patil, Taillie, and Talwalker (1993) discussed the use of weighteddistribution methods in encounter sampling and ecological studies. Iyengar and Zhao(1994) studied the impact of weight functions on M.L.E.’s of parameters of exponentialfamilies, and families with monotone likelihood ratio. In fisheries, Taillie, Patil, and

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Hennemuth (1995) modeled populations of fish stocks using weights. Jain and Nanda(1995) discussed the multivariate weighted distributions. Ahmad (1995) worked onthe estimation of multivariate kernel density for samples drawn from weighted distri-butions. Iyengar, Kvam, and Singh (1999) investigated the conditions under whichFisher information about parameter in weighted distribution is greater than the origi-nal density. Barmi and Simonoff (2000) estimated the baseline density on the basis ofa random sample from its weighted version. Navarro, Aguila, and Ruiz (2001) char-acterized probability distributions by considering the relationship between failure rateand mean residual life of a distribution and its weighted version. Oluyede and George(2002) established the inequalities, stochastic orderings, as well as useful ageing no-tions for weighted distributions. Gove (2003a) surveyed some of the applications ofsize biased distributions in forestry. Gove (2003b) considered Weibull distribution andestimated its parameters under length and size biased sampling. Navarro, Ruiz, andAguila (2003) discussed some methods to detect biased samples. Pakes et al. (2003)characterized one parameter exponential and power series families using the weighteddistributions. Nair and Sunoj (2003) identified a class of form-invariant continuousbivariate distribution under weighting and showed that there exists a natural conjugateprior for parameters of identified class. Chung and Kim (2004) worked on measuringrobustness for weighted distributions under Bayesian perspective. Sunoj and Sankaran(2005) studied the bivariate weighted models in the context of reliability modeling.Sunoj and Maya (2006) established some structural relationships between a variableand its weighted version in the context of maintainability function and reversed repairrate. Navarro, Ruiz, and Aguila (2006) studied some properties associated to multi-variate weighted distributions. Oluyede (2006) obtained error bounds for exponentialapproximations to the classes of weighted residual and equilibrium lifetime distribu-tions. Lele and Keim (2006) described how the estimation of weight function in thetheory of weighted distribution is closely connected with the estimation of resourceselection probability functions. Chakraborty and Das (2006) studied some propertiesof weighted version of quasi binomial distribution. Akman et al. (2007) proposed atest for detection of length-biased sampling. Oluyede and Terbeche (2007) studiedweighted distributions in the context of energy and expected uncertainty measures.Misra, Gupta, and Dhariyal (2008) provided the conditions under which the weightedversion of a random variable preserves the aging property and stochastic orders. Fur-man and Zitikis (2008) used the concept of weighted distributions in the context ofpremium calculation principles in actuarial sciences. Maya and Sunoj (2008) studiedsome dynamic information measures between a true distribution and a weighted dis-tribution and established relationships between these distributions. Blazej (2008) used

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1.5. Review of literature

the representation of weighted distributions to obtain some results concerning their re-lations with life distributions. Kim (2008) proposed a class of weighted multivariatenormal distributions. Kim (2008) introduced a class of weighted normal distributionswith normal, skew-normal distributions as its special cases and discussed its applica-tion. Alavi, Chinipardaz, and Rasekh (2008) discussed hypothesis in the context ofweighted distributions. Bartoszewicz (2009) established relations between weighteddistributions and classes of life distributions and stochastic orders. Alavi and Chinipar-daz (2009) obtained form-invariant multivariate normal distribution under a quadraticweight function. Kokonendji, Kiessé, and Balakrishnan (2009) studied semiparamet-ric estimation of kernel through weighted distributions. Leiva, Sanhueza, and Angulo(2009) derived the length biased version of Birnbaum-Saunders and showed its ade-quacy in modeling water quality. Shahbaz, Shahbaz, and Butt (2010) studied basicproperties of weighted Weibull distribution. Kumar, Taneja, and Srivastava (2010) in-troduced length biased version of weighted residual inaccuracy measures and derivedthe lower bound of weighted residual inaccuracy measure. Riabi, Borzadaran, andYari (2010) derived β -entropy for Pareto-type distribution and its weighted versions.Ghitany et al. (2011) proposed two-parameter of weighted Lindley distribution formodeling survival data. Ye, Oluyede, and Pararai (2012) studied the weighted versionof generalized beta distribution of second kind. Li, Yu, and Hu (2012) investigatedproperties of weighted distributions for some general weight functions. Kokonendjiand Casany (2012) used Radon-Nikodym theorem and showed that any count distribu-tion is more generally a weighted version of any other count distribution. Sunoj andSreejith (2012) describes the behavior of density of weighted models using the con-cept of reciprocal subtangent. Borgesa, Rodrigues, and Balakrishnan (2012) definedthe new classes of correlated weighted Poisson processes which is a generalizationof the class of weighted Poisson process defined by Balakrishnan and Kozubowski(2008). Subramani and Haridoss, 2013 used weighted Poisson distribution in singlesampling attributed plan in the context of quality control. Izadkhah, Roknabadi, andBorzadaran (2013) tried to provide conditions under which weighted distribution pre-serve the reversed mean residual life order. Rashwan (2013) studied the properties ofdouble weighted Rayleigh distribution and estimated its parameters. Ali (2013) workedon Bayesian estimation of the weighted Lindley distribution. Karimi and Alavi (2014)considered problems on statistical hypothesis testing under weighted sampling and ob-served the effect of weight function in obtaining the most powerful test. Izadkhaha,Roknabadia, and Borzadaran (2014) established results about preservation of decreas-ing mean residual life (m.r.l.) by weighted distributions after obtaining the conditionsunder which m.r.l. order is preserved. Kharazmi, Mahdavi, and Fathizadeh (2014)

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generalized weighted exponential distribution proposed by Gupta and Kundu (2009).Al-kadim and Hussein (2014) proposed Length-Biased version of weighted exponen-tial and Rayleigh distribution and showed their applications. Oluyede and Ye (2014)proposed weighted version of Degum distribution and compared it with weighted gen-eralized gamma, generalized gamma and generalized Lindley distribution. Al-Mutairi,Ghitany, and Kundu (2015) estimated the stress-strength parameter based on two in-dependent weighted Lindley random variables. Asgharzadeh et al. (2016) generalizedLindley distribution using the concept of weighted distribution and illustrated its appli-cation. Gupta and Arnold (2016) established the conditions under which monotonic-ity of hazard rate is preserved under weighting. Ghorbanpour, Chinipardaz, and Alavi(2018) made comparison between the Fisher information of weighted distributions andthe parent distributions. Behdani, Borzadaran, and Gildeh (2018) studied the relation-ships between probability distributions and their weighted versions in terms of inequitymeasures and draw some interesting connections.

In the last decade, researchers studied different weighted versions of various al-ready existing distributions see, for instance, Mahdy (2011), Das and Roy (2011), Shi,Oluyede, and Pararai (2012), Shakhatreh (2012), Roy and Adnan (2012), Mir, Reshi,and Ahmed (2013), Ahmed and Reshi (2013), Reshi, Ahmed, and Mir (2014b), Reshi,Ahmed, and Mir (2014c), Reshi, Ahmed, and Mir (2014a), Mahdavi (2015), Modi andGill (2015), and Mobarak, Nofal, and Mahdy (2017) etc.

1.6 Objectives of the study

Keeping in view the importance of weighted distribution in statistical modeling, we areinterested in showing adequacy of weighted versions of some already existing proba-bility models on assuming observations get recorded with probability proportional tosome weight function w(x). For illustration purpose, weighted Maxwell distribution,weighted transmuted power distribution, transmuted version of weighted exponentialdistribution and weighted gamma-Pareto distribution have been studied in detail and itis shown that introduced versions prove comparatively to be more adequate than theparent distributions for modeling some considered data sets.

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1.6. Objectives of the study

TAB

LE

1.3:

Diff

eren

tcha

ract

eris

tics

ofga

mm

a,ex

pone

ntia

l,M

axw

ell,

pow

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dPa

reto

dist

ribu

tion.

``

```

```

`C

hara

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ma

Exp

onen

tial

Max

wel

lPa

reto

Pow

erfu

nctio

n

Supp

ort

(0,∞

)(0,∞

)(0,∞

)[θ,∞

)(0,θ

]

Dis

trib

utio

nfu

nctio

nγ(α

,x/β

)

Γ(α

)1−

e−λ

xer

f( √ θ/2

x) −√ 2θ πx

exp(−

θx2 /

2)1−( θ x

) k( x θ

) αQ

uant

ilefu

nctio

nβ

Q−

1 (α,0,p)

−lo

g(1−

p)λ

√ 2Q−

1(3/2,1−

p)θ

θ

(1−

p)1/

kθ

p1 α,

p∈[0,1]

Haz

ard

rate

xα−

1 e−x/

β

βα

Γ(α

,x/β

)λ

2θx3/

2

2xθ+√

2πer

fc( √ θ

/2x) ex

p(θ

x2 2)

k xα

xα−

1

θα−

xα

m.g

.f.(1−

βt)−

α,

t<1/

βλ

λ−

t,t<

λ

( t2 θ+

1) exp(

t2 2θ) er

f(t

√2√

θ

) +1 +

√ 2π

θt

k(−

θt)

k Γ(−

k,−

θt),

t<0

α[Γ(α

)−

Γ(α

,−tθ)]

(−tθ)α

Cha

ract

eris

ticfu

nctio

n(1−

iβt)−

α,

t<1/

βλ

λ−

it,

t<λ

√ 2π

θit−

ι

( t2 θ−

1) exp( −t2 2θ

) erfi( t√

2θ

) −i

k(−

iθt)

k Γ(−

k,−

iθt),

t<0

α[Γ(α

)−

Γ(α

,−it

θ)]

(−it

θ)α

Mea

nα

βλ−

1

√ 8π

θ

θk

k−

1,k

>1

αθ

α+

1

Med

ian

βQ−

1( α

,0,

1 2

)λ−

1lo

g(2)

√ 2 θQ−

1( 3 2

,1 2

)21/

k θ2−

1/α

θ

Mod

eβ(α−

1),

α≥

10

√ 2 θθ

θ

Var

ianc

eα

β2

λ−

23π−

8π

θ

kθ2

(k−

1)2 (

k−

2),k

>2

αθ

2

(α+

1)2 (

α+

2)

Skew

ness

2 √α

22√

2(16−

5π)

(3π−

8)3/

22(

k+

1)k−

3

√ k−

2k

,k>

3−

2(α−

1)α+

3

√ α+

2α

Kur

tosi

s6 α+

36

π(1

6+

15π)−

192

(8−

3π)2

6(k3

+k2−

6k−

2)(k−

4)(k−

3)k

,k>

43( 3α

3+

5α2+

4)α(α

+3)(α

+4)

Ent

ropy

α+

log

βΓ(α

)+(1−

α)ψ

(α)

1−

log

λ

log(

2θ)+

γ−

log( √ 2/

πθ

3/2) −

1 2k+

1k

+lo

g( θ k

)lo

g( θ α

) +α−

1α

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Chapter 2Weighted Maxwell-BoltzmannDistribution

2.1 Introduction

Maxwell Boltzmann distribution commonly known as Maxwell Distribution (MD) iswidely used in Physics especially in statistical mechanics to describe the speed of aparticle in an idealized gas. This distribution can also be used to describe the dis-tribution of

√e1 + e2 + e3 where e1,e2 and e3 are the measurement errors in position

coordinates of a particle in a 3-dimensional space. This distribution was initially setforth in 1859 by the Scottish physicist James Clerk Maxwell. In 1871, Maxwell’sresult was generalized by a German physicist Ludwig Boltzmann to express the dis-tribution of energies among molecules. This distribution is often used to describe thespeed of a particle moving in a 3-dimensional space such that it’s movement alongthe three coordinate axes are independently and normally distributed random variableswith mean zero and variance equal to the inverse of rate parameter. However, insteadof 3-dimensional space if a particle moves in a 2-dimensional space it’s speed is betterdescribed by Rayleigh distribution. MD can be used to find the distribution of par-ticle’s kinetic energy (E) which is related to particle’s speed (v = x) by the formulaE = 1

2mx2, provided the distribution of speed is known. The p.d.f. of MD is given by

f (x) =

√( m2πkT

)34πx2 exp(−mx2

2kT), (2.1)

where m denotes the mass of particle, k the Boltzmann’s constant and T thermody-namic temperature.

? The content of this chapter has been published in "International Journal of Applied Mathematics& Information Sciences (Natural Sciences Publishing)", Vol. 12(1) pp. 193-202 (2018).

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Chapter 2. Weighted Maxwell-Boltzmann Distribution

Re-parameterizing equation (2.1) by m/kT = θ , we have the resulting p.d.f. as givenin (2.2).

f (x;θ) =

√2π

θ3/2x2 exp

(−θx2

2

),x≥ 0,θ > 0. (2.2)

In Physics and Chemistry there are many applications of MD. MD forms the basisof the kinetic energy of gases, which explains many fundamental properties of gases,including pressure and diffusion. This distribution is sometimes referred to as the dis-tribution of velocities, energy and magnitude of momenta of molecules. It was allabout the importance of MD in studying the randomness in physical and chemical sci-ences. In Statistics it can be tackled in a different way and one would like to analyzeand investigate its different statistical properties. Researchers considered MD to an-alyze and discuss its behavior in different perspectives like Tyagi and Bhattacharya(1989b) considered MD as a lifetime model for the first time and discussed the Baye’sand minimum variance unbiased estimation procedures for it’s parameter and relia-bility function see Tyagi and Bhattacharya (1989a). Empirical Baye’s estimation forthe MD was studied by Bekker and Roux (2005). Dey and Maiti (2010) carried outthe Bayesian estimation of parameters of MD under different loss functions. Kazmi,Aslam, and Ali (2012) studied the Bayesian estimation for two component mixture ofMD, assuming type-I censored data. Lu (2011) developed acceptance sampling plansfor MD after truncating the life test at a pre-fixed time, whereas Gui (2014) proposeda double acceptance sampling plan for MD based on the truncated life test. Tomer andPanwar (2015) carried out the estimation procedures for MD under type-I progressivehybrid censoring scheme. Huang and Chen (2016) studied tail behavior of general-ized MD. Dey et al. (2016) considered two parameter version of MD and estimated itslocation and scale parameter using both the frequentist and Bayesian approach.

Here we consider MD and are interested in introducing its weighted version onassuming the weight function w(x) = xω .

2.1.1 Derivation of weighted Maxwell-Boltzmann distribution

Consider the weight function w(x) = xω , where ω > 0 is the weight parameter. There-fore,

E[w(x)] =2ω/2+1Γ((ω +3)/2)√

πθ ω. (2.3)

Now, using the definition of weighted distribution given by (1.53), we get the p.d.f.of weighted Maxwell-Boltzmann distribution (WMD) and is given by (2.4).

fw (x;θ ,ω) =θ (ω+3)/2xω+2 exp

(−θx2/2

)2(ω+1)/2Γ((ω +3)/2)

, x > 0, θ > 0, ω > 0. (2.4)

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2.1. Introduction

We use X ∼WMD(θ ,ω) as the notation for denoting a random variable X follow-ing WMD with rate θ and weight parameter ω in the rest of chapter. C.d.f., reliabilityfunction and hazard rate of WMD are respectively given by (2.5), (2.6) and (2.7).

Fw (x;θ ,ω) = 1−Γ((ω +3)/2,θx2/2

)Γ((ω +3)/2)

. (2.5)

Rw (x;θ ,ω) =Γ((ω +3)/2,θx2/2

)Γ((ω +3)/2)

. (2.6)

hw (x;θ ,ω) =θ (ω+3)/2xω+2 exp

(−θx2/2

)2(ω+1)/2Γ((ω +3)/2,θx2/2)

, (2.7)

where∫

∞

a xm−1e−xdx = Γ(m, a) is upper incomplete gamma integral.

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

f ω( x

, θ,

ω)

ω = 0 (MD)θ = 1θ = 2

ω = 1 (LBMD)θ = 1θ = 2

ω = 2 (ABMD)θ = 1θ = 2

(a) p.d.f.

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

F ω(

x, θ

, ω)

ω = 0 (MD)θ = 1θ = 2

ω = 1 (LBMD)θ = 1θ = 2

ω = 2 (ABMD)θ = 1θ = 2

(b) c.d.f.

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

R ω(

x, θ

, ω)

ω = 0 (MD)θ = 1θ = 2

ω = 1 (LBMD)θ = 1θ = 2

ω = 2 (ABMD)θ = 1θ = 2

(c) reliability

0 1 2 3 4 5 6

01

23

45

6

x

h ω( x

, θ,

ω)

ω = 0 (MD)θ = 1θ = 2

ω = 1 (LBMD)θ = 1θ = 2

ω = 2 (ABMD)θ = 1θ = 2

(d) hazard rate

FIGURE 2.1: Density, c.d.f., reliability and hazard rate curves at differ-ent values of ω and θ .

It can be seen from Figure 2.1(a) that WMD is a positively skewed distributionwith strictly increasing hazard rate as shown by Figure 2.1(d). It can also be observedfrom Figure 2.1(d) that hazard rate increases on increasing θ and decreases with theincrease in ω .

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TABLE 2.1: Special cases of WMD at different values of ω .

weight parameter (ω) ω = 0 ω = 1 ω = 2

Distribution MD LBMD ABMD

fw(x;θ)

√2π

θ 3/2x2e−θx2/2 θ 2x3

2e−θx2/2 2−3/2

Γ(5/2)θ 5/2x4e−θx2/2

Fw(x;θ) 1− Γ(3/2,θx2/2)Γ(3/2)

1− Γ(2,θx2/2)Γ(2)

1− Γ(5/2,θx2/2)Γ(5/2)

2.2 Structural properties

In this section, various structural properties of WMD have been discussed. Expressionfor the rth moment about origin, mean, variance, coefficient of variation, skewness andkurtosis, m.g.f. and characteristic function are given.

Theorem 2.1. The rth moment about origin of a random variable X ∼WMD(θ ,ω) isgiven by

µ′r =

(2θ

)r/2Γ((ω + r+3)/2)

Γ((ω +3)/2),r = 1,2,3, ... (2.8)

Proof. By the definition of rth moment about origin we have

µ′r = E[xr]

µ′r =

∫∞

0xr fw (x;θ ,ω)dx

µ′r =

∫∞

0xr θ (ω+3)/2xω+2

2(ω+1)/2Γ((ω +3)/2)exp(−θx2/2

)dx

µ′r =

θ (ω+3)/2

2(ω+1)/2Γ((ω +3)/2)

∫∞

0xω+r+2 exp

(−θx2/2

)dx

µ′r =

(2θ

)r/2Γ((ω + r+3)/2)

Γ((ω +3)/2),r = 1,2,3, ...

First four moments about origin are respectively obtained by substituting r = 1,2,3,4in (2.8) and are given as follows:

µ′1 = µ =

√2θ

Γ(w4)

Γ(w3). (2.9)

µ′2 =

2Γ(w5)

θΓ(w3). (2.10)

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2.2. Structural properties

µ′3 =

(2θ

) 32 Γ(w6)

Γ(w3). (2.11)

µ′4 =

4Γ(w7)

θ 2Γ(w3), (2.12)

where wi =ω + i

2, i = 0,1,2,3,4...

Similarly, variance, coefficient of variation, skewness and kurtosis are respectivelygiven by (2.13), (2.14), (2.15) and (2.16).

σ2 =

2[Γ(w3)Γ(w5)−Γ(w4)2

]θ [Γ(w3)]

2 . (2.13)

cv =

√Γ(w3)Γ(w5)−Γ(w4)2

Γ(w4). (2.14)

γ1 =Γ(w3)2[

Γ(w3)Γ(w5)−Γ(w4)2]3/2

[Γ(w6)−3

Γ(w4)Γ(w5)

Γ(w3)+2Γ(w4)3

Γ(w3)2

].

(2.15)

γ2 =[Γ(w5)Γ(w3)−Γ(w4)2

]−2 [Γ(w7)Γ(w3)3−4Γ(w6)Γ(w4)Γ(w3)2

+12Γ(w5)Γ(w3)Γ(w4)2−6Γ(w4)4−3Γ(w5)Γ(w3)2]. (2.16)

Theorem 2.2. Square of sample coefficient of variation is asymptotically unbiased es-

timator of the square of population coefficient of variation, i.e., limn−→∞

E[

SnXn

]2=

(σ

µ

)2

,

where Xn and S2n are respectively the mean and variance of a sample.

Proof. Let X1,X2, ..,Xn, be a random sample of size n with mean Xn and variance S2n,

drawn from WMD(θ ,ω). Therefore,

E[Xn]= µ & var

[Xn]= σ

2/n. (2.17)

Also, E[Xn2] = var

[Xn]+[E

(Xn)]2. (2.18)

Using (2.17), (2.13) and (2.9) in (2.18), we get

E[Xn2] =

2[Γ(w3)Γ(w5)− (1−n)Γ(w4)2

]nθ [Γ(w3)]

2 . (2.19)

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Also, E[S2n] = σ

2 =2[Γ(w3)Γ(w5)−Γ(w4)2

]θ [Γ(w3)]

2 . (2.20)

Now, E[S2n] = E

[S2

n

Xn2 Xn

2

]= E

[S2

n

Xn2

]E[Xn

2].

Therefore, E

[S2

n

Xn2

]=

E[S2n]

E[Xn2].

Using (2.19) and (2.20), we obtain

E

[S2

n

Xn2

]=

Γ(w3)Γ(w5)−Γ(w4)2

1nΓ(w3)Γ(w5)−

(1n −1

)Γ(w4)2 .

Applying limn−→∞

on both sides, we get

limn−→∞

E

[S2

n

Xn2

]=

Γ(w3)Γ(w5)−Γ(w4)2

Γ(w4)2

limn−→∞

E

[S2

n

Xn2

]=

√

Γ(w3)Γ(w5)−Γ(w4)2

Γ(w4)

2

= (cv)2.

Theorem 2.3. The m.g.f. and characteristic function of X ∼WMD(θ ,ω) are respec-tively given by (2.21) and (2.22).

MX (t) =∞

∑r=0

tr

r!

(2θ

)r/2Γ((ω + r+3)/2)

Γ(w3), t ∈ R. (2.21)

ΦX (t) =∞

∑r=0

(it)r

r!

(2θ

)r/2Γ((ω + r+3)/2)

Γ(w3). (2.22)

Proof. From the definition of m.g.f. we have

MX (t) = E[etX]

MX (t) =∫

∞

0etx fw (x;θ ,ω)dx

MX (t) =∫

∞

0etx θ (ω+3)/2xω+2 exp

(−θx2/2

)2(ω+1)/2Γ((ω +3)/2)

dx

MX (t) =∞

∑r=0

tr

r!θ (ω+3)/2 ∫ ∞

0 xω+r+2 exp(−θx2/2

)dx

2(ω+1)/2Γ((ω +3)/2)

MX (t) =∞

∑r=0

tr

r!

(2θ

)r/2Γ((ω + r+3)/2)

Γ(w3).

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2.3. Entropy

Also, we know that ΦX (t) = MX (it) . Therefore,

ΦX (t) =∞

∑r=0

(it)r

r!

(2θ

)r/2Γ((ω + r+3)/2)

Γ(w3).

2.3 Entropy

Theorem 2.4. Renyi and Shannon entropy of WMD are respectively given by

HR (δ ) =1

1−δlog

[(2θ)(δ−1)/2Γ((2ω +2δ +1)/2)

δ (δω+2δ+1)/2 Γ((ω +3)/2)δ

]. (2.23)

HS =12[3+ω− log(2θ)+2logΓ((ω +3)/2)− (ω +2)Ψ((ω +3)/2)] . (2.24)

Proof. From the definition of Renyi entropy given by (1.34) in Section 1.2.6.1, we canwrite

HR (δ ) =1

1−δlog[∫

∞

0 fw (x;θ ,ω)δ dx

]

HR (δ ) =1

1−δlog

∫ ∞

0

θ (ω+3)/2xω+2 exp(−δθx2/2)

2(ω+1)/2Γ((ω +3)/2)

δ

dx

HR (δ ) =

11−δ

log

[θ δ (ω+3)/2 ∫ ∞

0 xδ (ω+2) exp(−δθx2/2)dx

2δ (ω+1)/2 Γ((ω +3)/2)δ

]

HR (δ ) =1

1−δlog

[(2θ)(δ−1)/2Γ((2ω +2δ +1)/2)

δ (δω+2δ+1)/2 Γ((ω +3)/2)δ

].

Similarly, using the definition of Shanon entropy given by (1.36), we get

HS = limδ−→1

11−δ

log

[(2θ)(δ−1)/2Γ((2ω +2δ +1)/2)

δ (δω+2δ+1)/2 Γ((ω +3)/2)δ

]HS =

12[3+ω− log(2θ)+2logΓ((ω +3)/2)− (ω +2)Ψ((ω +3)/2)] .

From Table 2.2, it is quite evident that on increasing the value of weight parameter(ω) for the fixed value of rate parameter (θ), mean, variance and entropy increase,whereas the coefficient of variation, skewness and kurtosis start decreasing. Similarly,on increasing the value of rate parameter for the fixed value of weight parameter, mean,variance and entropy decreases, whereas the other three characteristics, i.e., coefficientof variation, skewness and kurtosis remains unaffected due to their independence from

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TABLE 2.2: Characteristics of WMD at different values of θ and ω .

θ ω Mean σ2 cv γ1 γ2

Renyi Entropy HR(δ ) Shannonδ Entropy

0.5 0.7 0.9999 HS0 1.5958 0.4535 0.422 0.486 0.108 1.161 1.077 0.99618 0.99615

1 1 1.8799 0.4657 0.363 0.406 0.059 1.189 1.104 1.01927 1.019252 2.1277 0.4729 0.323 0.354 0.037 1.207 1.118 1.03182 1.031790 1.1284 0.2268 0.422 0.486 0.108 0.814 0.731 0.64960 0.64958

2 1 1.3293 0.2329 0.363 0.406 0.059 0.843 0.757 0.67269 0.672682 1.5045 0.2365 0.323 0.354 0.037 0.860 0.772 0.68525 0.685220 0.7137 0.0907 0.422 0.486 0.108 0.356 0.273 0.19146 0.19144

5 1 0.8408 0.0931 0.363 0.406 0.059 0.385 0.299 0.21455 0.214532 0.9515 0.0946 0.323 0.354 0.037 0.401 0.313 0.22710 0.22708

θ . It can also be seen from the last two columns of Table 2.2 that Renyi entropyapproaches to Shannon entropy as the order (δ ) of Renyi entropy tends to 1.

2.4 Bonferroni and Lorenz curve

Theorem 2.5. The Bonferroni and Lorenz curve associated with a random variableX ∼WMD(θ ,ω) are respectively given by (2.25) and (2.26).

B(p) =Γ(w4)−Γ

(w4, q2θ/2

)pΓ(w4)

. (2.25)

L(p) =Γ(w4)−Γ

(w4, q2θ/2

)Γ(w4)

, (2.26)

where q = F−1w (p;θ ,ω), p ∈ [0,1].

Proof. From the definition of Bonferroni curve given by (1.30) in Section 1.2.5.1, wecan write,

B(p) =1

pµ

∫ q

0x fw (x;θ ,ω)dx

B(p) =Γ((ω +3)/2)

p√

2/θΓ((ω +4)/2)

∫ q

0x

θ (ω+3)/2xω+2 exp(−θx2/2

)2(ω+1)/2Γ((ω +3)/2)

dx

B(p) =θ (ω+4)/2 ∫ q

0 xω+3 exp(−θx2/2

)dx

p 2(ω+2)/2Γ((ω +4)/2)

B(p) =Γ((ω +4)/2)−Γ

((ω +4)/2, q2θ/2

)pΓ((ω +4)/2)

B(p) =Γ(w4)−Γ

(w4, q2θ/2

)pΓ(w4)

.

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2.5. Order statistics

Similarly, on using (1.31) given in Section 1.2.5.1, we get

L(p) =Γ(w4)−Γ

(w4, q2θ/2

)Γ(w4)

.

2.5 Order statistics

Let x(1) ≤ x(2) ≤ ... ≤ x(r) ≤ ... ≤ x(n) be an ordered random sample of odd size (i.e.,n = 2m+ 1,m = 0,1,2, ...) from WMD(θ ,ω). Then, the p.d.f. of X(1), X(n), X(r) andX(m+1) (sample median) are respectively given by (2.27), (2.28), (2.29) and (2.30).

fX1:n(x) =nθ w3x2w2

Γ(w3,θx2/2

)n−1

2w2 Γ(w3)n exp(−θx2/2

). (2.27)

fXn:n(x) =nθ w3x2w2

Γ(w3)−Γ

(w3,θx2/2

)n−1

2w1 Γ(w3)n exp(−θx2/2

). (2.28)

fXr:n(x) =n!θ w3x2w2 Γ(w3)n−r

Γ(w3)−Γ(w3,θx2/2

)r−1

(r−1)!(n− r)!2w1 Γ(w3)n exp(−θx2/2

).

(2.29)

fXm+1:n(x) =(2m+1)!

m! m!θ w3x2w2 exp

(−θx2/2

)2w1 [Γ(w3)]

2m+1

[Γ(w3)

Γ(w3,θx2/2)−1]m

(2.30)

×[Γ(w3,θx2/2

)]2m.

2.6 Estimation of parameters

In this section parameters of WMD are estimated by using the method of maximumlikelihood estimation and method of moments. Fisher information matrix is also ob-tained and is used to obtain the 100(1−α)% asymptotic confidence interval of M.L.E.’s.

2.6.1 Maximum likelihood estimation

Let x1,x2, ...,xn be a random sample of size n from WMD(θ ,ω). Therefore, it’s likeli-hood function is given by (2.31).

l (Θ|x) =θ n(ω+3)/2

∏ni=1 xω+2

i

2n(ω+1)/2 Γ((ω +3)/2)n exp(−θ

2

n

∑i=1

x2i ), where Θ = (θ ,ω). (2.31)

Log likelihood function is given by

log[l (Θ|x)] = n(ω +3)2

logθ − n(ω +1)2

log2−n logΓ((ω +3)/2)

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+(ω +2)n

∑i=1

logxi−θ

2

n

∑i=1

x2i . (2.32)

Differentiating log likelihood function partially with respect to θ and ω we get thefollowing two gradients:

∂ log[l (Θ|x)]∂θ

=n(ω +3)

2θ−

n

∑i=1

x2i

2. (2.33)

∂ log[l (Θ|x)]∂ω

=n2

[logθ − log2−Ψ

(ω +3

2

)+

2n

n

∑i=1

logxi

]. (2.34)

On equating the derived gradients to zero and reducing them to simplified forms, weobtain following system of two equations:

θ =n(ω +3)∑

ni=1 x2

i. (2.35)

logθ − log2−Ψ

(ω +3

2

)+

2n

n

∑i=1

logxi = 0. (2.36)

Substituting (2.35) in (2.36), we get

log(

ω +32

)−ψ

(ω +3

2

)= log

n

∑i=1

x2i −

2n

n

∑i=1

logxi− logn. (2.37)

It is impossible to obtain the M.L.E. of ω by solving (2.37) manually for ω . There-fore, an estimate of ω is computed numerically by using the following code writtenin Wolfram Mathematica programming language after supplying a guess value sayω = ω0 and a data set say x.

FindRoot[Log[(\[Omega]+3)/2]-PolyGamma[0,(\[Omega]+3)/2]-Log[Total

[x^2]]+(2Total[Log[x]])/Length[x]+Log[Length[x]]==0,\[Omega],

Subscript[\[Omega],0]]

After obtaining the numerical estimate of ω say ωmle, it is substituted in (2.35) in orderto have the corresponding estimate of θ which is given by

θmle =n(ωmle +3)

∑ni=1 x2

i. (2.38)

Now, on using (1.45) given in Section 1.2.10.2.1, we get Fisher information matrixassociated with WMD(θ ,ω) and is given by

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2.6. Estimation of parameters

I(Θ) =−E

∂ 2 log l (Θ|x)

∂θ 2∂ 2 log l (Θ|x)

∂θ∂ω

∂ 2 log l (Θ|x)∂ω∂θ

∂ 2 log l (Θ|x)∂ω2

I(Θ) =

n(ω +3)

2θ 2−n2θ

−n2θ

n4

Ψ′(

ω +32

) , (2.39)

where Ψ(x) =∂ logΓ(z)

∂ z=

Γ′(z)

Γ(z)is known as digamma or Psi function. Thus, the

asymptotic 100(1−α)% confidence interval for Θ is given by

Θ ∈[

Θmle± zα/2

√diag

(I−1(Θmle)

)]Since, Θ = (θ ,ω) therefore we can write

θ ∈[

θmle± zα/2

√I−1

1,1(Θmle)

]and ω ∈

[ωmle± zα/2

√I−1

2,2(ωmle)

]. (2.40)

Therefore, 95% confidence interval for θ and ω is respectively given by

θ ∈[

θmle±1.96√

I−11,1(Θmle)

]and ω ∈

[ωmle±1.96

√I−1

2,2(ωmle)

](2.41)

where, z0.025 = 1.96 and I−1i, j represents the element belonging to ith row and jth column

of inverse of Fisher information matrix.

2.6.2 Method of moments

Let X1,X2, ..,Xn, be a random sample of size n with m1 = ∑ni=1

xi

nand m2 = ∑

ni=1

x2in

be the first two sample moments about origin. Therefore, on using the definition ofmoment estimator given in Section 1.2.10.1.2, we obtain the following system of twonon-linear equations:√

2θ

Γ((ω +4)/2)Γ((ω +3)/2)

=n

∑i=1

xi/n =⇒ θ = 2[

nΓ((ω +4)/2)Γ((ω +3)/2)∑

ni=1 xi

]2

(2.42)

2θ

Γ((ω +5)/2)Γ((ω +3)/2)

=n

∑i=1

x2i /n =⇒ θ =

2nΓ((ω +5)/2)Γ((ω +3)/2)∑

ni=1 x2

i(2.43)

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From (2.42)) and (2.43) we get[Γ((ω +4)/2)

∑ni=1 xi

]2

=Γ((ω +3)/2)Γ((ω +5)/2)

n∑ni=1 x2

i(2.44)

Equation (2.44) is nonlinear, solving it manually for ω is hectic. It is therefore,solved by using the function "FinRoot" in Wolfram Mathematica with initial guessvalue say ω = ω0 as:

FindRoot[(Gamma[(\[Omega]+4)/2]/Total[x])^2-(Gamma[(\[Omega]+3)/2]

Gamma[(\[Omega]+5)/2])/(Length[x]Total[x^2]),\[Omega],Subscript[

\[Omega],0]]

After obtaining the estimate of ω say ωmm, it is substituted for ω either in (2.42) or(2.43) so as to obtain the moment estimate of θ denoted by θmm.

2.7 Application

In this section, we have fitted WMD to four different types of data sets. The considereddata sets include three real life and a simulated one. Comparison is made betweenthe special cases of WMD in terms of distribution of best fit. Comparison criteria,viz., AIC, AICc, and BIC are used as tools for finding the model of best fit for theconsidered data sets.

2.7.1 Real life data

The R code for generating three real life data sets is as follows:

> install.packages("faraway")

> library(faraway)

> intensity <- star$light

> intensity

[1] 5.23 5.74 4.93 5.74 5.19 5.46 4.65 5.27 5.57 5.12 5.73 5.45

[13] 5.42 4.05 4.26 4.58 3.94 4.18 4.18 5.89 4.38 4.22 4.42 4.85

[25] 5.02 4.66 4.66 4.90 4.39 6.05 4.42 5.10 5.22 6.29 4.34 5.62

[37] 5.10 5.22 5.18 5.57 4.62 5.06 5.34 5.34 5.54 4.98 4.50

Dataset named "intensity" is related to the logarithm of light intensity of 47 stars inthe star cluster CYG OB1.

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2.7. Application

> noise<-resceram$noise

> noise

[1] 1.11 0.95 0.82 1.70 1.22 0.97 1.60 1.11 1.52 1.22 1.54 1.18

"noise" represents the current noise of four resistors mounted in a combination of 3on different crematic plates.

> wear<-abrasion$wear

> wear

[1] 235 236 218 268 251 241 227 229 234 273 274 226 195 270

[15] 230 225

Dataset titled as "wear" is a vector regarding the amount of wear recorded on feedingfour materials into a wear testing machine on using a Latin square design.

2.7.2 Simulation

Herein a random sample of size n from WMD with θ = θ1 (say) and ω = ω1 (say) isgenerated by using the inverse sampling method which is discussed in Section 1.2.7.1.Following the inverse sampling procedure, we obtained equation (2.45) on equatingthe c.d.f. of WMD to a number p.

Fw (x;θ = θ1,ω = ω1) = 1−Γ[(ω1 +3)/2,θ1x2/2

]Γ [(ω1 +3)/2]

= p,wherep∼ U(0,1). (2.45)

On solving the equation (2.45) for x, at n independently generated values of p, wewill obtain n independent values of x from WMD with θ = θ1 and ω = ω1. Theequation (2.45) is non linear in nature and can’t be solved manually for x. Hence, therequired solution is obtained by employing the function "uniroot" in R programming.The procedure for generation of random numbers from WMD by using the inversesampling method is given in form of following algorithm.

> DataWMD<-function(n,s,t,w)#n=sample size,s=seed,t=theta,w=omega

+ set.seed(s)

+ U=runif(n,0,1)

+ library(zipfR)

+ cdf<-function(x,t,w)

+ fn<-1-Igamma((w+3)/2,(t*x^2)/2,

+ lower=FALSE)/gamma((w+3)/2)

+ data=c() #Create an empty vector

+ for(i in 1:length(U))

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+ fn<-function(x)cdf(x,t,w)-U[i]

+ uni<-uniroot(fn,c(0,100000))

+ data=c(data,uni$root)

+ return(data)

Therefore, a random sample of size 100 from WMD with θ = 3 and ω = 3 is generatedas:

> simulated<-DataWMD(100,1,3,3)

> head(simulated)

[1] 1.09103 1.20576 1.41003 1.90537 1.01379 1.87957 2.02599

TABLE 2.3: M.L.E.’s, moment estimates, AIC, BIC and AICc.

Data Distn. M.L.E.’s Moment estimates −2log l AIC BIC AICcωmle θmle ωmm θmm

inte

nsity WMD 36.5575 1.55488 36.5734 1.55551 79.650 83.650 87.351 83.92

MD 0∗ 0.11794 0∗ 0.10137 161.83 163.83 165.68 163.9LBMD 1∗ 0.15726 1∗ 0.14069 148.24 150.24 152.09 150.3ABMD 2∗ 0.19657 2∗ 0.18021 138.18 140.18 142.03 140.3

nois

e WMD 7.95318 6.74692 7.79031 6.64660 2.4277 6.4277 7.3975 7.761MD 0∗ 1.84792 0∗ 1.64286 9.9411 11.941 12.426 12.34

LBMD 1∗ 2.46395 1∗ 2.28015 7.2943 9.2943 9.7792 9.694ABMD 2∗ 3.07983 2∗ 2.92065 5.5491 7.5491 8.0341 7.949

wea

r WMD 59.0587 0.00107 58.8915 0.00107 143.63 147.63 149.17 148.5MD 0∗ 0.00005 0∗ 0.00004 178.46 180.46 181.23 180.7

LBMD 1∗ 0.00007 1∗ 0.00006 173.68 175.68 176.46 175.9ABMD 2∗ 0.00009 2∗ 0.00008 170.29 172.29 173.07 172.6

sim

ulat

ed WMD 5.10605 4.05085 5.16938 4.08249 69.931 73.931 79.142 74.06MD 0∗ 1.49912 0∗ 1.35267 110.62 112.62 115.23 112.7

LBMD 1∗ 1.99891 1∗ 1.87739 92.002 94.002 96.608 94.04ABMD 2∗ 2.49872 2∗ 2.40475 80.893 82.893 85.498 82.93

After the fitting of WMD and its special cases to the considered data sets, maximumlikelihood estimates, moment estimates, −2log likelihood along with comparison cri-terion, viz., AIC, BIC and AICc are estimated and given in Table 2.3. From Table2.3 it is quite obvious that for all the considered data sets, it is WMD which proves tobe the model of best fit because of possessing the least values of AIC, AICc and BICfollowed by ABMD, LBMD and MD respectively.

From Table 2.4 it can be noticed that larger the Fisher information number corre-sponding to the parameter, smaller is the variance in it and narrower is its confidenceinterval. It is also clear from Figure 2.2.

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2.8. Conclusion

TABLE 2.4: 95% confidence interval for M.L.E.’s of WMD along withthe Fisher Information and covariance matrix.

DataI(Θmle) I−1(Θmle) 95% confidence interval

θ ω

θmle ωmle θmle ωmle lower upper lower upper

intensity 0.9235 2.1863 20.697 52.418θmle 384.506 -15.114 0.10378 2.57397ωmle -15.114 0.60934 2.57397 65.4840

noise 1.2565 12.237 0.5576 16.464θmle 1.44371 -0.8893 7.84688 11.6144ωmle -0.8893 0.60082 11.6144 18.8552

wear 0.0003 0.0018 16.283 101.83θmle 4.31e+08 -7455.73 1.447e-07 8.235e-03ωmle -7455.73 0.131009 8.235e-03 4.763e+02

simulated 2.9013 5.2004 2.9453 7.2668θmle 24.6994 -12.343 0.34400 0.60736ωmle -12.343 6.99102 0.60736 1.21537

2.8 Conclusion

In this Chapter, various properties associated with WMD have been studied and dis-cussed in detail. Three real life data sets, viz., intensity, noise, wear and a sim-ulated one is considered for illustrating the validity of WMD in statistical modeling.Simulated data set is generated by employing the inverse sampling method. After thefitting of WMD to the considered data sets, maximum likelihood estimates, momentestimates of parameters along with Fisher information matrix and its inverse is esti-mated. Different statistical measures like AIC, BIC and AICc have also been computedfor the special cases of WMD. As we know that the distribution with lowest AIC, BICand AICc is considered to be the distribution of best fit. It is observed that WMD pos-sesses the least AIC, BIC and AICc followed respectively by ABMD, LBMD and MD.Hence, it can be concluded that WMD proves to be more flexible and the distributionof best fit for the considered data sets in comparison to its special cases.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

θ

dens

ity

(a) intensity

20 30 40 50

0.00

0.01

0.02

0.03

0.04

0.05

ω

dens

ity

(b) intensity

0 5 10 15

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

θ

dens

ity

(c) noise

0 5 10 15 20

0.00

0.02

0.04

0.06

0.08

ω

dens

ity

(d) noise

0.000 0.002 0.004 0.006 0.008 0.010

020

040

060

080

010

00

θ

dens

ity

(e) wear

20 40 60 80 100

0.00

50.

010

0.01

5

ω

dens

ity

(f) wear

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

θ

dens

ity

(g) simulated data

0 2 4 6 8 10

0.0

0.1

0.2

0.3

ω

dens

ity

(h) simulated data

FIGURE 2.2: Asymptotic normal curves of M.L.E.’s.

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2.8. Conclusion

0 2 4 6 8

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

intensity

D e

n s

i t y

EmpiricalWMDMDLBMDABMD

(a) intensity

2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

intensity

Pr.(X

≤x)


(b) intensity

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.5

1.0

1.5

noise

D e

n s

i t y


(c) noise

0.0 0.5 1.0 1.5 2.0

0.0

0.2

0.4

0.6

0.8

1.0

noise

Pr.(X

≤x)


(d) noise

50 100 150 200 250 300 350

0.00

00.

005

0.01

00.

015

0.02

0

wear

D e

n s

i t y


(e) wear

50 100 150 200 250 300 350

0.0

0.2

0.4

0.6

0.8

1.0

wear

Pr.(X

≤x)


(f) wear

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

simulated

D e

n s

i t y


(g) simulated data

0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

simulated

Pr.(X

≤x)


(h) simulated data

FIGURE 2.3: Empirical density and distribution curves along with thefitted ones.

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Chapter 3Weighted Transmuted Power FunctionDistribution

3.1 Introduction

Like exponential and Weibull distribution, power function distribution is intrigue innature and appealing, due to its flexibility in modeling various types of data sets. In alife accelerated testing problem, if the effect of an environmental stress under investi-gation is such that the hazard rate is expected not to be constant, the simple alternativeto exponential distribution may be offered by power function distribution, see Meni-coni and Barry (1996). Meniconi and Barry (1996) demonstrated the applicabilityof power function distribution in life testing and showed it to be a simple model forassessing reliability of some electric components. Properties of power function distri-bution has been extensively analyzed by various authors, not only in its baseline formbut with a variety of its new extensions see for instance, Cordeiro and Brito (2012)studied beta-power distribution after employing the concept of beta-G, suggested byEugene, Lee, and Famoye (2002). Haq et al. (2016) studied the transmuted version ofpower function distribution. Okorie et al. (2017) proposed a modified version of Powerfunction distribution, by using the idea of Marshall Olkin-G family of distribution dueto Marshell and Olkin (1997). Mutairi (2017) introduced transmuted weighted powerdistribution and discussed some of its properties and applications.

In this chapter, a four parameter extension of power function distribution is intro-duced by using quadratic rank transmutation map (QRTM) and concept of weighteddistribution. Different statistical properties of the introduced extension are analyzed.

? The content of this chapter has been published in "Revista de Investigacion Operacional", Vol.39(4) pp. 626-638 (2018).

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Chapter 3. Weighted Transmuted Power Function Distribution

The introduced extension is termed as weighted transmuted power function distributionand abbreviated as WTPFD.

3.1.1 Derivation of weighted transmuted power function distribu-tion

The p.d.f. and c.d.f. of a random variable X ∼ power(θ ,α) are respectively given by(3.1) and (3.2).

f (x;α,θ) =αxα−1

θ α,0 < x≤ θ ,θ ,α > 0. (3.1)

F(x;α,θ) =( x

θ

)α

,0 < x≤ θ ,θ ,α > 0. (3.2)

Before moving further to derive WTPFD, it is important to mention that powerfunction distribution is form invariant under the weight function w(x) = xω which isshown as below in Theorem 3.1.

Theorem 3.1. Power function distribution is form invariant under weight functionw(x) = xω .

Proof. From the definition of weighted distribution, weighted version of power func-tion distribution assuming weight function w(x) = xω is given by

f (x;α, θ , ω) =xω f (x;α, θ)

E[xω ]

f (x;α, θ , ω) =xω

αxα−1

θ α

αθ ω

α +ω

f (x;α, θ , ω) =(α +ω) xα+ω−1

θ α+ω.

Which is again the density of power function distribution with shape (α+ω) and samescale θ .

As a consequences of Theorem 3.1, it is not possible to derive a vast family ofdistributions on considering power function as a parent distribution. Thus, we firsttransmuted the power function distribution using QRTM and then on the assumingweight function w(x) = xω we get WTPFD the derivation of which is given as follows:

On using equation (1.12) and (1.13) given in Section 1.2.2.1.3 with G(x)=F(x;θ ,α)

and g(x) = f (x;θ ,α), we obtain c.d.f. and p.d.f. of transmuted version of X which arerespectively given by (3.3) and (3.4).

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3.1. Introduction

Gt(x;α,θ ,β ) =(1+β )(θx)α −βx2α

θ 2α,0≤ x≤ θ ,θ ,α > 0, |β | ≤ 1. (3.3)

gt(x;α,θ ,β ) =αθ α(1+β )xα−1−2βαx2α−1

θ 2α,0≤ x≤ θ ,θ ,α > 0, |β | ≤ 1. (3.4)

Now, on considering the weight function w(x) = xω and using the definition ofweighted distribution given by (1.52), we get p.d.f. of WTPD which is given by (3.5).

fw(x;α,θ ,β ,ω) =xωgt(x;α,θ ,β )

E [xω ],0≤ x≤ θ ,θ ,α,ω > 0, |β | ≤ 1.

fw(x;α,θ ,β ,ω) =(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα

θ 2α+ω(2α +ω−βω). (3.5)

Therefore, the c.d.f. of WTPFD is given by (3.6).

Fw(x;α,θ ,β ,ω) =xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)

θ 2α+ω(2α +ω−βω), (3.6)

0≤ x≤ θ ,θ ,α,ω > 0, |β | ≤ 1.

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

f ω(x

, α, θ

, β, ω

)

ω = 0−−−−−−(TPD)β = − 0.8β = 0β = 0.8

ω = 1−−−−−−(LBTPD)β = − 0.8β = 0β = 0.8

ω = 2−−−−−−(ABTPD)β = − 0.8β = 0β = 0.8

(a) p.d.f. plots

0 2 4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

x

F ω(x

, α, θ

, β, ω

)

ω = 0−−−−−−(TPD)β = − 0.8β = 0β = 0.8

ω = 1−−−−−−(LBTPD)β = − 0.8β = 0β = 0.8

ω = 2−−−−−−(ABTPD)β = − 0.8β = 0β = 0.8

(b) c.d.f. plots

FIGURE 3.1: P.d.f. and c.d.f. curves of WTPFD at different values ofθ , α, β and ω .

From now onwards, notation X ∼WT PFD(α,θ ,β ,ω) is used to denote a randomvariable X following WTPFD.

Theorem 3.2. Size biased version of a random variable X ∼ uniform(0,θ) is powerfunction distribution with shape (ω +1) and scale θ , where ω is the order of size bias.

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TABLE 3.1: Special Cases of WTPFD

Distribution Name Substitution Density function

TPFD ω = 0 αθ−2α xα−1 θ α(1+β )−2βxα

WPFD β = 0 (α +ω)θ−(α+ω)xα+ω−1

PFD ω = 0,β = 0 αθ−α xα−1

LBTPFD ω = 1(α +1)(2α +1)xα θ α(1+β )−2βxα

θ 2α+1(2α +1−β )

ABTPFD ω = 2(α +2)(α +1)xα+1 θ α(1+β )−2βxα

θ 2α+2(α +1−β )

WTUD α = 1(1+ω)(2+ω)xω θ(1+β )−2βx

θ 2+ω(2+ω−βω)

TUD ω = 0,α = 1 θ−1(1+β )−2βθ−2x

LBTUD ω = 1,α = 16xθ(1+β )−2βx

θ 3(3−β )

ABTUD ω = 2, α = 1(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα

θ 2α+ω(2α +ω−βω)

LBPFD ω = 1,β = 0 (α +1)θ−(α+1)xα

ABPFD ω = 2,β = 0 (α +2)θ−(α+2)xα+1

WUD β = 0,α = 1 (ω +1)θ−(ω+1)xω

UDω = 0,β = 0,α = 1

1θ

(or) α = 0,ω = 1

LBUD ω = 1,β = 0,α = 1 2θ−2x

ABUD ω = 2,β = 0,α = 1 3θ−3x2

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3.2. Characterization

Proof. P.d.f. of X ∼ uniform(0,θ) is given by f (x;θ) =1θ

. Therefore, the size biasedversion of X ∼ uniform(0,θ) is given by

f (x;θ , ω) =xω f (x;θ)

E[xω ]=

(ω +1) xω

θ ω+1 ,

which is the p.d.f. of power function distribution with shape (ω +1) and scale θ .

3.2 Characterization

Theorem 3.3. WTPFD with β ∈ [−1,0] is a mixture of two power function randomvariables with shape parameters α1 = α +ω and α2 = 2α +ω .

Proof. The distribution function of WTPFD given in (3.6) can be rewritten as:

Fw(x;α,θ ,β ,ω) =(1+β )(2α +ω)

2α +ω−βωFPF(x,α1,θ)−

2β (α +ω)

2α +ω−βωFPF(x,α2,θ),

(3.7)where FPF(x,α1,θ) and FPF(x,α2,θ) are respectively the distribution functions of twopower function random variables with shape parameters α1 = α +ω and α2 = 2α +ω .Rewriting (3.7) as follows:

Fw(x;α,θ ,β ,ω) = p1 FPF(x,α1,θ)+ p2 FPF(x,α2,θ) (3.8)

Now, to prove the distribution function given by (3.8) is a mixture of two powerfunction random variables, it is imperative to show that:

i) ∑ pi = 1, i = 1,2.

ii) pi ∈ [0,1], i = 1,2.

i) ∑ pi = p1 + p2 =(1+β )(2α +ω)

2α +ω−βω− 2β (α +ω)

2α +ω−βω= 1.

ii) Since, the numerator and denominator of p1 are both positive for all the possiblevalues of α, ω and β . Therefore, the only condition to be imposed on p1 such thatp1 ∈ [0,1] is

β : (1+β )(2α +ω)≤ 2α +ω−βω =⇒ β : β ≤ 0 =⇒ β ∈ [−1,0] (3.9)

Similarly, for p2 to lie in [0,1], following restriction must be imposed on parameter β .

β :−2β (α +ω)≤ 2α +ω−βω∩β : β ≤ 0 (3.10)

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=⇒ β : β ≥−1∩β ≤ 0 =⇒ β ∈ [−1,0]

From (3.9) and (3.10), it can be observed that the same restriction should be im-posed on parameter β in order to have p1 ∈ [0,1] and p2 ∈ [0,1].Therefore, for β ∈ [−1,0] , p1 and p2 satisfy (ii) also p1 + p2 = 1. Hence, it is con-cluded that for β ∈ [−1,0], WTPFD is the mixture distribution of two power functionrandom variables.

Theorem 3.4. For fixed value of α, ω and θ , the density of WTPFD is independentof β iff

x = θ

(α +ω

2α +ω

)1/α

.

Proof. Firstly to show for fixed value of α, ω and θ the density of WTPFD is inde-

pendent of parameter β at x = θ

(α +ω

2α +ω

)1/α

we follow as:

Substitute x = θ

(α +ω

2α +ω

)1/α

in the density of WTPFD, we get

fw

(x = θ

(α +ω

2α +ω

)1/α

;α,θ ,β ,ω

)=

α +ω

θ

(α +ω

2α +ω

)(α+ω−1)/α

,

which is independent of parameter β . Conversely, let β1,β2 be two different and arbi-trary values of β . Let the density of WTPFD is independent of β for the fixed valuesα,ω and θ , then

fw(x;α,θ ,β1,ω) = fw(x;α,θ ,β2,ω)

θ α(1+β1)−2β1xα

(2α +ω−β1ω)=

θ α(1+β2)−2β2xα

(2α +ω−β2ω)

x = θ

(α +ω

2α +ω

)1/α

.

Theorem 3.4 is also justified through Figure 3.2. Figure 3.2 represents density plotswith α = 2,θ = 10 and ω = 2 at different values of β . It is quite clear that the densityis independent of β at x = 8.164966 and the corresponding density comes out to be0.2177324.

Theorem 3.5. Density of WTPFD is:

(i) Increasing ∀ x ∈ [0, θ) if

(a) α +ω > 1 , β <α +ω−1

3α +ω−1.

(b) α +ω = 1 , β < 0.

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3.2. Characterization

0 2 4 6 8 10

0.00.1

0.20.3

0.40.5

0.6

x

f w(x, α

=2, θ

=10,

β=… ,

ω=2

)

density = 0.2177324

x = 8.164966

− 1 ≤ β < 0β = 00 < β ≤ 1

FIGURE 3.2: Density plots at different values of β .

(ii) Decreasing ∀ x ∈ [0, θ ] if α +ω = 1 , β > 0.

(iii) Increasing and decreasing if α +ω > 1 , β ≥ α +ω−13α +ω−1

.

(a) Increasing ∀ x ∈ [0, xg).

(b) Decreasing ∀ x ∈ (xg,θ ].

(c) Maximum at x = xg, where xg = θ

((1+β )(α +ω−1)

2β (2α +ω−1)

)1/α

.

(iv) Neither increasing nor decreasing if α +ω = 1 and β = 0.

Proof. Differentiating (3.5) w.r.t. x we get

f′w(x;α,θ ,β ,ω) =

(α +ω)(2α +ω)xα+ω−2

θ α+ω(2α +ω−βω)

((1+β )(α +ω−1)−2β (2α +ω−1)

( xθ

)α).

(3.11)

Following are the cases which are useful in understanding whether f′w(x;α,θ ,β ,ω) is

(<>=) 0 for different values of β .

i) α +ω > 1.

ii) α +ω = 1.

i) For f′w(x;α,θ ,β ,ω)> 0 given that α +ω > 1, it is must to have

(1+β )(α +ω−1)−2β (2α +ω−1)( x

θ

)α

> 0.

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x < θ

((1+β )(α +ω−1)

2β (2α +ω−1)

)1/α

.

Therefore, density is increasing ∀ x ∈ [0,θ ] if

((1+β )(α +ω−1)

2β (2α +ω−1)

)1/α

> 1, i.e., β <α +ω−1

3α +ω−1.

However, if ((1+β )(α +ω−1)

2β (2α +ω−1)

)1/α

≤ 1, i.e., β ≥ α +ω−13α +ω−1

.

Then, ∃! a point xg ∈ [0,θ ] such that

f′w(x;α,θ ,β ,ω)

> 0 ∀x ∈ [0,xg).

= 0 x = xg

< 0 ∀x ∈ (xg,θ ].

where, xg = θ

((1+β )(α +ω−1)

2β (2α +ω−1)

)1/α

.

Hence, for β ≥ α +ω−13α +ω−1

given that α +ω > 1, the density is increasing ∀x ∈[0,xg), decreasing ∀x ∈ (xg,θ ] and attains maximum at x = xg.ii) Given that α +ω = 1.

f′w(x;α,θ ,β ,ω)> 0 if 2β

( xθ

)α

< 0, i.e., β < 0 and

f′w(x;α,θ ,β ,ω)< 0 if 2β

( xθ

)α

> 0, i.e., β > 0.

Therefore, given that α +ω = 1, density is increasing if β < 0 and decreasing if β > 0.However, if β = 0, then the density is neither increasing nor decreasing.


In this Section, various structural properties of WTPFD are investigated. Mathematicalexpression for rth moment about origin, variance, m.g.f., characteristic function havebeen derived.

Theorem 3.6. rth moment about origin of random variable X ∼WT PFD(α,θ ,β ,ω)

is given by

µ′r = E [xr] =

θ r(α +ω)(2α +ω)2α− (β −1)(ω + r)(r+α +ω)(r+2α +ω)(2α +ω−βω)

, r = 1,2,3, ... (3.12)

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Proof. We know that

µ′r =

∫θ

0xr fw(x;α,θ ,β ,ω)dx.

µ′r =

∫θ

0xr (α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα

θ 2α+ω(2α +ω−βω)dx.

After simplifying the expression, we get

µ′r =

θ r(α +ω)(2α +ω)2α− (β −1)(ω + r)(r+α +ω)(r+2α +ω)(2α +ω−βω)

Substituting r = 1 in (3.12), we obtain Mean of WTPFD and is given by (3.13)

µ′1 =

θ(α +ω)(2α +ω)2α− (β −1)(ω +1)(1+α +ω)(1+2α +ω)(2α +ω−βω)

(3.13)

Similarly, variance of WTPFD is given by (3.14)

σ2 =

θ 2(α +ω)(2α +ω)2α− (β −1)(ω +2)(2+α +ω)(2+2α +ω)(2α +ω−βω)

−[

θ(α +ω)(2α +ω)2α− (β −1)(ω +1)(1+α +ω)(1+2α +ω)(2α +ω−βω)

]2

(3.14)

Mathematical expression for higher order moments like coefficient of skewnessand kurtosis are not obtained in simplest form but have been computed numerically atdifferent values of θ , α, β , ω and are reported in Table 3.2.

Theorem 3.7. The moment generating function and characteristic function of a ran-dom variable X ∼WT PFD(α,θ ,β ,ω) are respectively given by

MX (t) =(α +ω)(2α +ω)

[θ α(1+β )∑

∞j=0

t jθ α+ω+ j

j!(α +ω + j)−2β ∑

∞j=0

t jθ 2α+ω+ j

j!(2α +ω + j)

]θ 2α+ω(2α +ω−βω)

.

(3.15)

ΦX (t) =(α +ω)(2α +ω)

[θ α(1+β )∑

∞j=0

(it) jθ α+ω+ j

j!(α +ω + j)−2β ∑

∞j=0

(it) jθ 2α+ω+ j

j!(2α +ω + j)

]θ 2α+ω(2α +ω−βω)

.

(3.16)

Proof. The m.g.f. of X ∼WT PFD(α,θ ,β ,ω) is given by

MX (t) = E[etX] .

MX (t) =∫

∞

0etx fw(x;α,θ ,β ,ω)dx.

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MX (t) =∫

θ

0etx (α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα


On simplifying the integral, we obtain.

MX (t) =(α +ω)(2α +ω)


[θ

α(1+β )∞

∑j=0

t jθ α+ω+ j

j!(α +ω + j)−2β

∞

∑j=0

t jθ 2α+ω+ j

j!(2α +ω + j)

].

Similarly, characteristic function is easily obtained by using the following relation:

ΦX (t) = MX (it)

ΦX (t) =(α +ω)(2α +ω)


[θ

α(1+β )∞

∑j=0

(it) jθ α+ω+ j

j!(α +ω + j)−2β

∞

∑j=0

(it) jθ 2α+ω+ j

j!(2α +ω + j)

].

3.4 Reliability measures

Reliability function and hazard rate associated with WTPFD are respectively given by(3.17) and (3.18).

R(t) = 1− tα+ω θ α(1+β )(2α +ω)−2tαβ (α +ω)θ 2α+ω(2α +ω−βω)

. (3.17)

h(t) =(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα

θ 2α+ω(2α +ω−βω)− tα+ω θ α(1+β )(2α +ω)−2tαβ (α +ω).

(3.18)

Theorem 3.8. Mean residual life of a random variable T ∼WT PFD(α,θ ,β ,ω) isgiven by

m(t) =(θ − t)−

[θ α(1+β )(2α +ω)(θ α+ω+1− tα+ω+1)

θ 2α+ω(2α +ω−βω)(α +ω +1)− 2β (α +ω)(θ α+1− tα+1)

θ 2α+ω(2α +ω−βω)(α +1)

]1− tα+ω θ α(1+β )(2α +ω)−2tαβ (α +ω)


.

(3.19)

Proof. From the definition of mean residual life (m.r.l.) given by (1.21) in Section1.2.3.5, we can write

m(t) =∫

∞

t R(x)dxR(t)

.

m(t) =

∫θ

t

[1− xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)


]dx

1− tα+ω θ α(1+β )(2α +ω)−2tαβ (α +ω)θ 2α+ω(2α +ω−βω)

.

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3.5. Entropy measure

m(t) =(θ − t)−

[θ α(1+β )(2α +ω)(θ α+ω+1− tα+ω+1)

θ 2α+ω(2α +ω−βω)(α +ω +1)− 2β (α +ω)(θ α+1− tα+1)

θ 2α+ω(2α +ω−βω)(α +1)

]1− tα+ω θ α(1+β )(2α +ω)−2tαβ (α +ω)


.

0 2 4 6 8 10

0.0

0.5

1.0

1.5

T

h(t |

θ=

10)

β = 0.5α = 5 , ω = 0α = 5 , ω = 1α = 5 , ω = 2

β = − 0.5α = 5 , ω = 0α = 5 , ω = 1α = 5 , ω = 2

(a) hazard curve

0 2 4 6 8 10

02

46

810

T

m (t

| θ

=10

)

β = 0.5α = 5 , ω = 0α = 5 , ω = 1α = 5 , ω = 2

β = − 0.5α = 5 , ω = 0α = 5 , ω = 1α = 5 , ω = 2

(b) m.r.l. curve

FIGURE 3.3: Hazard rate and m.r.l. at different values of θ ,α,β and ω .

From Figure 3.3, it can be observed that WTPFD possesses increasing hazard rate(IFR) and decreasing mean residual life (DMRL). Therefore, according to the chain ofimplications existing between different ageing properties, DMRL implies NBUE (newbetter than used in expectation, i.e., µ(0)≥ µ(t),∀t ≥ 0) see Deshpande, Kochar, andSingh (1986) and Kochar and Wiens (1987). Thus WTPFD belongs to NBUE family.WTPFD belonging to NBUE family is also justified by the result of Hall and Wellner(1984) which states that for NBUE family, coefficient of variation has to be ≤ 1. FromTable 3.2, it is quite clear that cv is ≤ 1 for WTPFD.

3.5 Entropy measure

Theorem 3.9. Renyi entropy of order δ associated with X ∼WT PFD(α,θ ,β ,ω) isgiven by

HR (δ ) =1

1−δlog

[(α +ω)(2α +ω)


δ

(3.20)

×∞

∑k=0

(δ

k

)θ α(1+β )δ−k (−2β )kθ δ (α+ω−1)+αk+1

δ (α +ω−1)+αk+1

].

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Proof. From the definition of Renyi entropy given by (1.34) in Section 1.2.6.1 ofChapter 1, we write

HR (δ ) =1

1−δlog[∫

∞

0 f (x,α,θ ,β ,ω)δ dx

], δ ≥ 0 & δ 6= 1

HR (δ ) =1

1−δlog

[∫θ

0

(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα


δ

dx

]

HR (δ ) =1

1−δlog

[(α +ω)(2α +ω)


δ ∫ θ

0xδ (α+ω−1) θ α(1+β )−2βxαδ dx

].

Therefore, on using Newton’s generalized binomial theorem we obtain.

HR (δ ) =1

1−δlog

[(α +ω)(2α +ω)


δ

×∞

∑k=0

(δ

k

)θ α(1+β )δ−k (−2β )kθ δ (α+ω−1)+αk+1

δ (α +ω−1)+αk+1

].

Shanon entropy (HS) which is the limiting form of Renyi entropy (i.e., HS =

limδ−→1

HR (δ ) ) is not obtained in the form of mathematical expression but has been

computed numerically and is given in Table 3.2.

TABLE 3.2: Characteristics of WTPFD at different values of parame-ters.

Parameters Mean Variance C.V. Skewness Kurtosis HR(δ ) Shannonδ Entropy

θ α β ω µ σ2 cv γ1 γ2 0.3 0.6 0.9999 HS1 69.697 505.969 0.3227 -0.7024 -0.338 4.531 4.421 4.3433 4.3429

1 -0.3 2 76.957 338.563 0.2391 -0.9589 0.3752 4.370 4.228 4.1028 4.10271 62.963 591.221 0.3862 -0.3983 -0.815 4.551 4.512 4.4740 4.4740

0.3 2 72.353 412.111 0.2806 -0.7209 -0.219 4.439 4.339 4.2524 4.25241 92.840 49.4668 0.0758 -1.7809 4.1204 3.582 3.224 2.9680 2.9679

10 -0.3 2 93.373 42.5092 0.0698 -1.8041 4.2983 3.515 3.148 2.8907 2.89071 90.459 65.4479 0.0894 -1.3670 2.2390 3.706 3.427 3.2394 3.2394

100 0.3 2 91.182 56.9231 0.0827 -1.3962 2.3798 3.643 3.355 3.1624 3.16231 96.115 15.7516 0.0413 -1.9858 5.5227 3.055 2.639 2.3566 2.3565

20 -0.3 2 96.279 14.4331 0.0395 -1.9923 5.5890 3.014 2.596 2.3136 2.31351 94.784 21.1319 0.0485 -1.5558 3.2763 3.187 2.854 2.6418 2.6417

0.3 2 95.007 19.4876 0.0465 -1.5648 3.3255 3.148 2.812 2.5986 2.59861 97.334 7.63775 0.0284 -2.0677 6.1439 2.710 2.272 1.9797 1.9796

30 -0.8 2 97.412 7.18319 0.0275 -2.0702 6.1772 2.681 2.242 1.9499 1.94981 96.412 10.2977 0.0333 -1.6313 3.7399 2.846 2.491 2.2697 2.2696

0.8 2 96.518 9.72715 0.0323 -1.6353 3.7629 2.818 2.462 2.2397 2.2397

From Table 3.2, it can be seen that the coefficient of variation (cv) is ≤ 1 justifyingthat WTPFD belongs to NBUE family, see Hall and Wellner (1984). From column (8)of the same table it is clear that WTPFD is negatively skewed distribution which is alsoapparent from density plots given in Figure 3.1(a). The last two columns of Table 3.2justify that Shannon entropy (HS) is the limiting case of Renyi entropy (HR(δ )) as theorder (δ ) of HR(δ ) tends to 1. It is also observed that with the increase in ω and α ,

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mean, coefficient of kurtosis increase whereas variance, coefficient of variation, skew-ness and entropy decrease. Increase in β leads to decline in mean and kurtosis whilevariance, coefficient of variation, skewness and entropy increase. It is also apparentthat Renyi entropy decreases with the increase in order (δ ).


Let x(1)≤ x(2)≤ ...≤ x(r)≤ ...≤ x(n) be an ordered random sample of odd size (i.e., n=2m+ 1,m = 0,1,2, ...) from WT PFD(α,θ ,β ,ω). Therefore, on using (1.23), (1.24),(1.25) and (1.28) given in Section 1.2.4 of Chapter 1, we get the p.d.f. of X(r),X(1),X(n)

and X(m+1), and are respectively given by (3.21), (3.22), (3.23) and (3.24).

fXr:n(x) =n!

(r−1)!(n− r)!

[xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)


]r−1

×[1− xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)


]n−r

×

(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxαθ 2α+ω(2α +ω−βω)

. (3.21)

fX1:n(x) = n[

1− xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)θ 2α+ω(2α +ω−βω)

]n−1

×


. (3.22)

fXn:n(x) = n[

xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)θ 2α+ω(2α +ω−βω)

]n−1

×


. (3.23)

fXm+1:n(x) =(2m+1)!

m!m!

[1− xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)


]m

×[xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)


]m

×


. (3.24)

In Figure 3.4 density of rth order statistics from WT PFD(θ ,α,β ,ω) is plotted atdifferent values of r with n = 11,θ = 10,α = 5,β = 0.5,ω = 2. These density plotsdepict that increase in order (r) leads to decrease in dispersion of order statistics. Den-sity curve corresponding to r = 6 is of sample median whereas, density correspondingto r = 11 is of nth order statistics which is same as that of exponentiated version ofWTPFD.

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0 2 4 6 8 10

0.00.5

1.01.5

2.0

X(r)

f X r:11

(x)fX1:11

(x) −−−Ist Order

fX4:11(x)

fX6:11(x) −−−Median

fX8:11(x)

fX11:11(x) − − nthOrder (or) Exponentiated WTPFD

FIGURE 3.4: Density of rth order statistics at θ = 10,α = 5,β =0.5,ω = 2 and n = 11.


Theorem 3.10. The Bonferroni and Lorenz curve of a random variable X ∼WT PFD(α,θ ,β ,ω) are respectively given by

B(p) =(ω +α +1)(ω +2α +1)

p2α− (β −1)(ω +1)θ ω+2α+1

[θ α(1+β )qα+ω+1

α +ω +1− 2βq2α+ω+1

2α +ω +1

]. (3.25)

L(p) =(ω +α +1)(ω +2α +1)

θ ω+2α+12α− (β −1)(ω +1)

[θ α(1+β )qα+ω+1

α +ω +1− 2βq2α+ω+1

2α +ω +1

]. (3.26)

where, p ∈ [0,1], q = F−1(p)

Proof. From the definition of Bonferroni curve given by (1.30) in Section 1.2.5.1 ofChapter 1, we can write

B(p) =1

pµ

∫ q

0x fω(x;α,θ ,β ,ω)dx.

B(p) =1

pµ

∫ q

0x(α +ω)(2α +ω)xα+ω−1 θ α(1+β )−2βxα


B(p) =(ω +α +1)(ω +2α +1)

p2α− (β −1)(ω +1)θ ω+2α+1

[θ α(1+β )qα+ω+1

α +ω +1− 2βq2α+ω+1

2α +ω +1

].

Therefore, Lorenz curve is given by

L(p) = pB(p) =(ω +α +1)(ω +2α +1)

2α− (β −1)(ω +1)θ ω+2α+1

[θ α(1+β )qα+ω+1

α +ω +1− 2βq2α+ω+1

2α +ω +1

].

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3.8. Random number generation

3.8 Random number generation

Random numbers from the WTPFD are generated by using the inverse sampling method,discussed in Section 1.2.7.1 of Chapter 1. Using the inverse sampling method, weobtained the following equation on equating the c.d.f. of WTPFD to a number p ∼uniform(0,1).

Fω(x;α,θ ,β ,ω) =xα+ω θ α(1+β )(2α +ω)−2xαβ (α +ω)

θ 2α+ω(2α +ω−βω)= p. (3.27)

Solving (3.27) for x, at n independent values of p, we get a required sample of sizen from WTPFD. It is very tedious to solve (3.27) manually because of being non-linear.Thus, the general algorithm for generation of random numbers from WTPFD writtenin R programming is given as:

> DataWTPFD<-function(n,s,t,a,b,w)

+ set.seed(s)

+ U=runif(n,0,1)

+ library(zipfR)

+ cdf<- function(x,t,a,b,w)

+ fn<-(x^(a+w)*(t^a*(1+b)*(2*a+w)-2*x^a*b*(a+w)))/(t^(2*a+w)*

+(2*a+w-b*w))



+ fn<-function(x)cdf(x,t,a,b,w)-U[i]

+ uni<-uniroot(fn,c(0,10),lower=0,upper=1000)


+ return(data)

3.9 Maximum likelihood estimation

Let 0≤ x(1)≤ x(2)≤ ...≤ x(n)≤ θ be an ordered sample of size n from WT PFD(α,θ ,β ,

ω). Therefore, its log likelihood function will be given by

log[l (α,θ ,β ,ω|x)] =n

∑i=1

log((β +1)θ α −2βxαi )+(α +ω−1)

n

∑i=1

logxi +n log(α +ω)

−n log(2α−βω +ω)−n(2α +ω) logθ +n log(2α +ω). (3.28)

Differentiating (3.28) w.r.t. θ we get the following gradient:

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0 20 40 60 80 100

−5−4

−3−2

−10

θ

Grad

ient o

f Log

likeli

hood

w.r.

t. θ

FIGURE 3.5: Gradient of log likelihood function w.r.t. θ at differentvalues of α,β and ω .

∂ log[l (α,θ ,β ,ω|x)]∂θ

=n

∑i=1

(β +1)α(β +1)θ α −2βxα

i− n(2α +ω)

θ. (3.29)

From Figure 3.5, it is clear that the gradient given by (3.29) is negative for all possiblevalues of θ , which implies that (3.28) is a decreasing function w.r.t. θ . Therefore,M.L.E. of θ under the restriction x(n) ≤ θ is given by θmle = x(n), whereas, M.L.E.’sfor the rest of three parameters ω, α and β are obtained by solving simultaneously theequation (3.30), (3.31) and (3.32). The following system of equations is obtained byequating the gradients of (3.28) w.r.t. ω, α and β to zero.

n

∑i=1

(4βxα

i − (β +1)xα

(n)

)(logx(n)− logxi)

(β +1)xα

(n)−2βxαi

− 2n2α−ω +ωβ

+n(4α +3ω)

(α +ω)(2α +ω)= 0.

(3.30)n

∑i=1

2β (2α +ω)xαi − (β +1)xα

(n)(α +ω)

(β +1)xα

(n)−2βxαi

= 0. (3.31)

n

∑i=1

xα

(n)(α +ω)− (2α +ω)xαi

(β +1)xα

(n)−2βxαi

= 0. (3.32)

The above system of equations is non-linear. Therefore, to obtain the estimates inclosed form is very tedious. Thus, R programming is used to find the required estimatesof parameters after the fitting of WTPFD to some considered data sets.

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3.10. Application

3.10 Application

This section is devoted to the application of WTPFD in describing a random data. Toshow the adequacy of WTPFD, it is fitted to two real life and a simulated data setalong with its special cases. After the fitting of WTPFD to the considered data sets,comparison is made between special cases of WTPFD in terms distribution of best fit.


Two real life data sets considered are light intensity and disposable income. Vari-able "light" and "dpi" are respectively extracted from parent data sets "star" and"savings" present in "faraway" package in R software, and named as "light_intensity" and "disposable_income" see, Faraway (2016). Dataset "star" consists oftwo columns, one related to log of the surface temperature and another regarding thelog of light intensity of 47 stars in the star cluster "CYG OB1". The dataset "savings"consists of 50 observations on 5 variables and "dpi" is one of them giving the dispos-able income in dollars ($) of 50 countries.

> install.packages("faraway")

> library(faraway)

> help(star) # Description about star dataset

> light_intesity<-star$light

> head(light_intesity)

[1] 5.23 5.74 4.93 5.74 5.19 5.46 4.65 5.27 5.57 5.12 5.73 5.45

> help(savings) # Description about savings dataset

> disposable_income<-savings$dpi

> head(disposable_income)

[1] 2329.68 1507.99 2108.47 189.13 728.47 2982.88 662.86

3.10.2 Simulated data

A sample of size n = 100 from WTPD with α = 5, θ = 10, β = −0.7 and ω = 2 isgenerated by using the algorithm given Section 3.8 . This generated data set is namedas "simulated_data" and is give as:

> simulated_data<-DataWTPFD(100,1,10,2,-0.5,4)

> head(simulated_data)

[1] 8.298195 8.705786 9.251606 9.867045 7.978410 9.852132 9.921221

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TABLE 3.3: M.L.E.’s, AIC and KS-test statistic.

DataDist

n. M.L.E.’s −2log l AIC D P-valueθmle αmle βmle ωmle

sim

ulat

edda

ta

WTPFD 9.989 5.503 -0.664 2.07e-6 214.48 222.48 0.0683 0.7415ABTPFD 9.989 4.264 -0.555 2∗ 215.86 221.86 0.0680 0.7435LBTPFD 9.989 4.888 -0.611 1∗ 215.15 221.15 0.0645 0.8002

TPFD 9.989 5.502 -0.664 0∗ 214.48 220.48 0.0607 0.8551SBPFD 9.989 4.239 0∗ 3.7858 218.87 222.87 0.0904 0.3875

PFD 9.989 8.025 0∗ 0∗ 218.87 222.87 0.0904 0.3874SBUD 9.989 1∗ 0∗ 7.0250 218.87 222.87 0.0904 0.3874

UD 9.989 1∗ 0∗ 0∗ 265.39 267.43 0.6559 2.2e-16

light

inte

nsity

WTPFD 6.29 0.014 0.999 8.2718 90.472 98.472 0.1483 0.2431ABTPFD 6.29 4.495 0.931 2∗ 91.642 97.642 0.1437 0.2865LBTPFD 6.29 5.290 0.921 1∗ 92.031 98.031 0.1456 0.2723

TPFD 6.29 6.107 0.911 0∗ 92.463 98.463 0.1478 0.2557PFD 6.29 4.282 0∗ 0∗ 108.19 111.00 0.2603 0.0034UD 6.29 1∗ 0∗ 0∗ 109.09 111.09 0.6264 2.22e-16

disp

osab

lein

com

e WTPFD 4001.89 0.012 0.997 1.0698 799.18 806.18 0.1293 0.3437ABTPFD 4001.89 1.62e-8 1 2∗ 839.90 845.90 0.3712 4.97e-7LBTPFD 4001.89 0.057 0.987 1∗ 799.18 805.18 0.1286 0.3497

TPFD 4001.89 0.809 0.859 0∗ 800.61 806.61 0.1331 0.3104PFD 4001.89 0.574 0∗ 0∗ 810.74 814.74 0.1833 0.0609UD 4001.89 1∗ 0∗ 0∗ 1171.7 1173.7 0.3824 4.38e-7

Observation with * as superscript refer to known quantities

Comparison criterion AIC and Kolmogorov-Smirnov test statistic (D) along with it’scorresponding P-value is computed numerically for each of the special case after theirfitting to the above mentioned data sets and are reported in Table 3.3.

Comparing different models in terms of fitting, using the statistical tools like AICand Kolmogorov-Smirnov test we know that the model with minimum AIC, D-Statisticor with maximum P-value is considered the model of best fit. Therefore on using thesame interpretation, Table 3.3 reveals that TPFD proves to be model of best fit forsimulated data set followed by LBTPFD, ABTPFD, PFD and UD. It is also observedthat SBPFD, PFD and SBUD provide the same result on their fitting to simulated data,justifying that PD is form-invariant and UD reduces to PD under size bias sampling.Due to the form-invariant property of PD under size bias sampling, only PD is includedfor comparison whereas the other two cases (i.e., SBPFD, SBUD) are dropped for reallife data sets. For light intensity, it is observed that ABTPFD proves to be the modelof best fit followed by LBTPFD, TPFD, WTPFD, PFD and UD. Similarly, for thedisposable income it is WTPFD which proves to be the distribution of best fit followedby TPFD, PFD, ABTPFD and UD respectively.

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3.10. Application

4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

simulated data

Den

sity

EmpiricalWTPDABTPDLBTPDTPDSBPD, PD, SBUDUD

(a) simulated data

4 6 8 10

0.0

0.2

0.4

0.6

0.8

1.0

simulated data

Pr.(

X≤

x)


(b) simulated data

4 5 6 7

0.0

0.1

0.2

0.3

0.4

0.5

0.6

light intensity

Den

sity


(c) light intensity

4.0 4.5 5.0 5.5 6.0 6.5

0.0

0.2

0.4

0.6

0.8

1.0

light intensity

Pr.(

X≤

x)


(d) light intensity

−1000 0 1000 2000 3000 4000 5000

0e+0

01e

−04

2e−0

43e

−04

4e−0

45e

−04

disposable income

Den

sity


(e) disposable income

0 1000 2000 3000 4000

0.0

0.2

0.4

0.6

0.8

1.0

disposable income

Pr.(

X≤

x)


(f) disposable income

FIGURE 3.6: Empirical along with fitted density and distributioncurves.

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3.11 Conclusion

An extension of power function distribution named as weighted transmuted powerfunction distribution is introduced and its different statistical properties are investi-gated and studied. Range of transmutation parameter for which WTPFD becomesmixture of two power function random variables is obtained. Point at which the densityis independent of transmutation parameter is found for fixed values of the remainingthree parameters. Mathematical expression for various statistical measures are derived.Parameters are estimated by using method of maximum likelihood estimation. The ad-equacy of WTPFD in expressing a random phenomenon, is supported and validatedthrough its fitting to a simulated and two real life data sets. Inverse sampling method isused to generate the random numbers from WTPFD. After the fitting of some specialcases of WTPFD to the considered data sets, it is concluded that TPFD, ABTPFD andWTPFD proves to be the model of best fit respectively for the simulated data, lightintensity and disposable income whereas, UD proves to be of worst fit for all of thethree considered data sets. It is observed that for the considered data sets mostly thenew cases proved to be best fit rather than the baseline distribution, i.e., power functiondistribution.

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Chapter 4Transmuted Weighted ExponentialDistribution

4.1 Introduction

Exponential distribution which is known for its constant hazard rate and possessing thememoryless property, is widely employed in modeling lifetime distributions. Lack ofmemory property is a critical part in most of the statistical analysis, that is why expo-nential distribution finds extensive applications in different areas especially in survivalanalysis. Exponential distribution has been used in different areas not only in its base-line form but with a variety of new extensions. The first extension of exponential wasdue to W. Weibull (1951) which is now known as Weibull distribution. Attempts togeneralize the exponential distributions lead to the formulation of various new distri-butions, e.g., Generalized exponential distribution by Gupta and Kundu (1999), betaexponential distribution by Nadarajah and Kotz (2005), weighted exponential distribu-tion by Gupta and Kundu (2009), Kumaraswamy exponential distribution by Cordeiroand Castro (2011) and exponentiated exponential distribution by Gupta (2001). Betageneralized exponential distribution was proposed and studied by Souza, Santos, andCordeiro (2010), gamma exponentiated exponential distribution by Ristic and Balakr-ishnan (2012b).

In this chapter, we have introduced a three parameter extension of exponentialdistribution by employing quadratic rank transmutation map (QRTM) and the conceptof weighted distribution. The weight function used is w(x) = xω . The developedextension is named as transmuted weighted exponential distribution and abbreviatedas TWED. The derivation of TWED is given below.

? The content of this chapter has been published in "Journal of Statistics Applications & Probability(Natural Sciences Publishing)", Vol. 6(1) pp. 219-232 (2017).

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Chapter 4. Transmuted Weighted Exponential Distribution

TABLE 4.1: Some special cases of TWED.

β ω Distn. ft (x) Ft (x)

00 ED λe−λx γ (1,λx)1 LBED λ 2xe−λx γ (2,λx)(1+βΓ(2,λx))2 ABED λ 3x2e−λx/2 γ (3,λx)(2+βΓ(3,λx))/4

6= 00 TED λe−λx

(1−β +2βe−λx

)γ (1,λx)(1+βΓ(1,λx))

1 TLBED λ 2e−λxx(1−β +2βΓ(2,λx)) γ (2,λx)(1+βΓ(2,λx))2 TABED λ 3e−λxx2 (2(1−β )+2βΓ(3,λx))/4 γ (3,λx)(2+βΓ(3,λx))/4

4.1.1 Derivation of transmuted weighted exponential distribution

Consider the weight function w(x) = xω and on using the definition of weighted dis-tribution given by (1.52) in Section 1.3, we get the p.d.f. of weighted exponentialdistribution (WED) which is given by

fw (x;ω,λ ) =xωλe−λx∫

∞

0 xωλe−λxdx=

λ (ω+1)xωe−λx

ω!, x≥ 0,λ > 0,ω > 0. (4.1)

The corresponding c.d.f. is given by

Fw (x;ω,λ ) = 1− Γ(ω +1,λx)Γ(ω +1)

=γ (ω +1,λx)

Γ(ω +1), x≥ 0,λ > 0,ω > 0. (4.2)

Now, on employing the definition of QRTM after treating the weighted exponen-tial as a baseline distribution, we obtain the p.d.f. and c.d.f. of transmuted weightedexponential distribution (TWED) which are respectively given by (4.3) and (4.4).

ft (x;ω,λ ,β ) =λ ω+1e−λxxω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 ,x≥ 0, |β | ≤ 1.

(4.3)

Ft (x;ω,λ ,β ) =γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

(Γ(ω +1))2 ,x≥ 0, |β | ≤ 1, (4.4)

where∫ b

0ta−1e−tdt = γ(a,b) is lower incomplete gamma intergral,∫

∞

bta−1e−tdt = Γ(a,b) is upper incomplete gamma integral and∫

∞

0ta−1e−tdt = Γ(a) = complete gamma integral.

Notation X ∼ TWED(x;ω,λ ,β ) is used to denote a random variable X followingtransmuted weighted exponential distribution from now onwards in this chapter.

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In this section, various structural properties of TWED are discussed. Mathematicalexpressions for rth moment about origin, m.g.f., characteristic function, mean and vari-ance are given.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0

0.5

1.0

1.5

2.0

2.5

x

f t(x, ω

, λ, β

)

λ = 5, ω = 0, β = 0 EDλ = 5, ω = 1, β = 0 LBEDλ = 5, ω = 2, β = 0 ABEDλ = 5, ω = 0, β = 0.5 TEDλ = 5, ω = 1, β = 0.5 TLBEDλ = 5, ω = 2, β = 0.5 TABED

(a) p.d.f

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.

00.

20.

40.

60.

81.

0

x

F t(x

, ω, λ

, β)


(b) c.d.f

FIGURE 4.1: P.d.f. and c.d.f. plots at different values of ω, λ and β .

Theorem 4.1. The rth moment about origin of a random variable X ∼ TWED(ω,λ ,β )

is given by

µ′r =

1λ r

[(1−β )Γ(ω + r+1)

Γ(ω +1)+

2βΓ(2ω + r+2) 2F1 (ω + r+1,2ω + r+2;ω + r+2;−1)

(ω + r+1)(Γ(ω +1))2

],

=1

λ r [(1−β )Gr +βFr] ,r = 1,2,3, ..., (4.5)

where Gr =Γ(ω + r+1)

Γ(ω +1),Fr =

2Γ(2ω + r+2) 2F1 (ω + r+1,2ω + r+2;ω + r+2;−1)

Γ(ω +1)2 (ω + r+1),

2F1 (a,b;c;z) =Γ(c)

Γ(b)Γ(c−b)

∫ 1

0

tb−1(1− t)c−b−1

(1− tz)a dt is hyper-geometric function and

2F1 (a,b;c;z) = 2F1 (a,b;c;z)Γ(c)

is regularized hyper-geometric function.

Proof. rth moment about origin is given by

µ′r = E[xr]

µ′r =

∫∞

0xr λ ω+1e−λxxω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

[Γ(ω +1)]2dx

µ′r =

λ ω+1

(Γ(ω +1))2 [(1−β )Γ(ω +1) I1 +2β I2] , (4.6)

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where I1 =∫

∞

0 xω+re−λxdx =Γ(ω + r+1)

λ ω+r+1 and I2 is of the form:

∫∞

0xre−sx

Γ(t,ux)dx =Γ(r+ t +1)2F1 (r+1,r+ t +1;r+2;−s/u)

(r+1) ur+1 . (4.7)

Therefore,

I2 =∫

∞

0xω+re−λx

Γ(ω +1,λx)dx

=Γ(2+ r+2ω)2 F1 (ω + r+1,2ω + r+2;ω + r+2;−1)

(ω + r+1)λ ω+r+1 .

Now, using I1 and I2 in (4.6) we get

µ′r =

1λ r

[(1−β )Γ(ω + r+1)

Γ(ω +1)+

2βΓ(2ω + r+2) 2F1 (ω + r+1,2ω + r+2;ω + r+2;−1)

(ω + r+1)(Γ(ω +1))2

]

=1

λ r [(1−β )Gr +βFr] .

Range of 2F1 (a,b;c;z) is R but the range of 2F1 (ω + r+1,2ω + r+2;ω + r+2;−1)is R+ which is obvious from Figure 4.2. Therefore, µ ′r ∈R+ for all the possible valuesof ω, λ , β and r.

FIGURE 4.2: 3D plot of hyper-geometric function at different values ofr and ω .

First four moments about origin are respectively obtained by substituting r = 1,2,3,4in (4.5) and are given as follows:

µ′1 = λ

−1 ((1−β )G1 +βF1) . (4.8)

µ′2 = λ

−2 ((1−β )G2 +βF2) . (4.9)

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µ′3 = λ

−3 ((1−β )G3 +βF3) . (4.10)

µ′4 = λ

−4 ((1−β )G4 +βF4) . (4.11)

and variance is given by

σ2 =λ

−2[(1−β )G2 +βF2− ((1−β )G1 +βF1)

2]. (4.12)

Descriptive measures like coefficient of variation (cv), skewness (γ1) and kurtosis(γ2) are not given in the closed form but are computed numerically and are given inTable 4.2.

TABLE 4.2: Characteristics of TWED at different value of ω, λ and β .

Distn. ω β λ µ σ2 γ1 γ2 cvHR(δ )

δ = 0.5 δ = 0.9999

TED 0

0.5 5 0.1500 0.0275 2.467 9.471 1.106 -0.4288 -0.905210 0.0750 0.0069 2.467 9.471 1.106 -1.1324 -1.5922

0 5 0.2000 0.0400 2.000 6.000 1.000 -0.2196 -0.606810 0.1000 0.0100 2.000 6.000 1.000 -0.9210 -1.2967

-0.5 5 0.2500 0.0475 1.715 4.504 0.872 -0.0880 -0.401610 0.1250 0.0119 1.715 4.504 0.872 -0.7784 -1.0962

TLBED 1

0.5 5 0.3250 0.0594 1.713 4.658 0.749 0.0761 -0.230810 0.1625 0.0148 1.713 4.658 0.749 -0.5722 -0.9197

0 5 0.4000 0.0800 1.414 3.000 0.707 0.2319 -0.029410 0.2000 0.0200 1.414 3.000 0.707 -0.4659 -0.7255

-0.5 5 0.4750 0.0894 1.215 2.309 0.629 0.3181 0.075510 0.2375 0.0223 1.215 2.309 0.629 -0.3762 -0.6137

TABED 2

0.5 5 0.5062 0.0925 1.394 3.141 0.601 0.3349 0.078110 0.2531 0.0231 1.394 3.141 0.601 -0.3482 -0.6136

0 5 0.6000 0.1200 1.155 2.000 0.577 0.4583 0.238810 0.3000 0.0300 1.155 2.000 0.577 -0.2163 -0.4499

-0.5 5 0.6938 0.1300 0.987 1.558 0.519 0.5389 0.313110 0.3469 0.0325 0.987 1.558 0.519 -0.1624 -0.3816

From Table 4.2, it can be observed that

• with the increase in weight parameter, mean, variance and entropy increasewhereas the coefficient of variation, skewness and kurtosis decrease.

• with the increase in transmutation parameter, mean, variance and entropy de-crease whereas the coefficient of variation, skewness and kurtosis increase.

• with the increase in rate parameter, mean, variance and entropy decrease whereasthe coefficient of variation, skewness and kurtosis remain constant.

• increase in order (δ ) leads to decrease in Renyi’s entropy (HR(δ )).

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Theorem 4.2. The moment generating function and characteristic function of X ∼TWED(ω,λ ,β ) are respectively given by

MX (t) =(1−β )

(1− t/λ )ω+1 +2βΓ(2ω +2)2F1 (ω +1,2ω +2,ω +2, t/λ −1)

Γ(ω +1). (4.13)

ΦX (t) =(1−β )

(1− it/λ )ω+1 +2βΓ(2ω +2)2F1 (ω +1,2ω +2,ω +2, it/λ −1)

Γ(ω +1). (4.14)

Proof. From the definition of m.g.f. we can write

MX (t) = E[etX ]

MX (t) =∫

∞

0etx ft (x)dx

MX (t) =∫

∞

0etx λ ω+1e−λxxω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 dx

MX (t) =λ ω+1

(Γ(ω +1))2

∫∞

0xω

[e−(λ−t)x(1−β )Γ(ω +1)+2βΓ(ω +1,λx)

]dx

MX (t) =λ ω+1

(Γ(ω +1))2

[(1−β )Γ(ω +1)

∫∞

0xωe−(λ−t)xdx+2β

∫∞

0xωe−(λ−t)x

Γ(ω +1,λx)dx].

Using the integral given in (4.7), we get

MX (t) =λ ω+1

(Γ(ω +1))2

[(1−β )

(Γ(ω +1))2

(λ − t)ω+1 +2βΓ(2ω +2)2F1 (ω +1,2ω +2,ω +2, t/λ −1)

(ω +1)λ ω+1

]MX (t) =

(1−β )

(1− t/λ )ω+1 +2βΓ(2ω +2)2F1 (ω +1,2ω +2,ω +2, t/λ −1)

Γ(ω +1).

Therefore, characteristic function is given by

ΦX (t) = MX (it) =(1−β )

(1− it/λ )ω+1 +2βΓ(2ω +2)2F1 (ω +1,2ω +2,ω +2, it/λ −1)

Γ(ω +1).

4.3 Reliability measures

Reliability function of TWED is given by

R(t) = 1− γ (ω +1,λ t) [Γ(ω +1)+βΓ(ω +1,λ t)]

(Γ(ω +1))2 . (4.15)

Hazard rate is given by

h(t) =λe−λ t (λ t)ω [(1−β )Γ(ω +1)+2βΓ(ω +1,λ t)]

(Γ(ω +1))2− γ (ω +1,λ t) [Γ(ω +1)+βΓ(ω +1,λ t)]. (4.16)

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4.3. Reliability measures

Theorem 4.3. The m.r.l. of a life time T ∼ TWED(ω,λ ,β ) is given by

m(t) =β ∑

ωj=0

Γ(2 j+1,2λ t)

( j!)2λ22 j+1

+β ∑ω

i6= j=0Γ(i+ j+1,2λ t)

i! j!λ2i+ j+1 − (β −1)∑ωj=0

Γ( j+1,λ t)j!λ

e−λ t ∑ωj=0

(λ t) j

j!

(βe−λ t ∑

ωj=0

(λ t) j

j!−β +1

)(4.17)

Proof. To obtain m.r.l. associated with TWED, we make use of alternate mathematicalexpression of reliability function which is obtained by using the following to identi-ties.

γ(n+1,y) = n!

[1− e−y

n

∑j=0

y j

j!

], n = 0,1,2, ... (4.18)

Γ(n+1,y) = n!e−yn

∑j=0

y j

j!, n = 0,1,2, ... (4.19)

Using (4.18) and (4.19) with n = ω and y = λx in (4.15), we obtained the alternatemathematical expression for reliability function which is given by

R(t) = e−λ tω

∑j=0

(λ t) j

j!

(βe−λ t

ω

∑j=0

(λ t) j

j!−β +1

).

Therefore, from the definition of m.r.l. we can write

m(t) =

∫∞

t e−λx∑

ωj=0

(λx) j

j!

(βe−λx

∑ωj=0

(λx) j

j!−β +1

)dx

e−λ t ∑ωj=0

(λ t) j

j!

(βe−λ t ∑

ωj=0

(λ t) j

j!−β +1

)

m(t) =β ∑

ωj=0∫

∞

te−2λx(λx)2 j

( j!)2 dx+β ∑ω

i6= j=0∫

∞

te−2λx(λx)i+ j

i! j!dx− (β −1)∑

ωj=0∫

∞

te−λx(λx) j

j!dx

e−λ t ∑ωj=0

(λ t) j

j!

(βe−λ t ∑

ωj=0

(λ t) j

j!−β +1

)

m(t) =β ∑

ωj=0

Γ(2 j+1,2λ t)

( j!)2λ22 j+1

+β ∑ω

i 6= j=0Γ(i+ j+1,2λ t)

i! j!λ2i+ j+1 − (β −1)∑ωj=0

Γ( j+1,λ t)j!λ

e−λ t ∑ωj=0

(λ t) j

j!

(βe−λ t ∑

ωj=0

(λ t) j

j!−β +1

) .

From Figure 4.3(a) it can be seen that hazard rate of TWED is increasing and be-comes asymptotically parallel to the hazard of exponential distribution which is con-stant and equal to rate parameter λ . Also from Figure 4.3(b) it can be observed thatm.r.l. associated with TWED is of unique nature. Initially it is constant and at aninstant increases abruptly with a sharp steep to a point after which it again remainsconstant.

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

01

23

45

6

T

h t(x

, ω, λ

, β)


(a) Hazard rate

0 2 4 6 8 10

020

4060

8010

0

T

µ (t)


(b) m.r.l.

FIGURE 4.3: Hazard rate and mean residual life function at differentvalues of ω, λ and β .

4.4 Renyi entropy

It is not possible to obtain Renyi entropy associated with TWED in closed form. There-fore, the reliable procedure for finding it’s numerical approximation is to employ theMonte Carlo integration technique discussed in Section 1.2.8. From the definition ofRenyi entropy given by (1.34) in Section 1.2.6.1, we can write

HR (δ ) =1

1−δlog[∫

∞

0[ ft (x;ω,λ ,β )]δ dx

]HR (δ ) =

11−δ

log[∫

∞

0[ ft (x;ω,λ ,β )]δ−1 ft (x;ω,λ ,β )dx

]HR (δ ) =

11−δ

log[∫

∞

0¯h(x;ω,λ ,β ) ft (x;ω,λ ,β )dx

]HR (δ ) =

11−δ

log [E[ ¯h(x;ω,λ ,β )]] , (4.20)

where

¯h(x;ω,λ ,β ) =

[λ ω+1e−λxxω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2

]δ−1

. (4.21)

Therefore, the numerical approximation of (4.20) can be obtained by using the follow-ing R program:

RenyiTWED=function(n,d,w,b,L)

R1=function(d,w,b,L)

D=rtwed(1,w,b,L)#generate a random number from TWED

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h=(L*exp(-L*D)*((L*D)^w)*((1-b)*gamma(w+1)+2*b*Igamma

(w+1,L*D,lower=F))/((gamma(w+1))^2))^(d-1)

hi=replicate(n,R1(d,w,b,L))

Integral=mean(hi)

renyi=(1/(1-d))*log(Integral)

return(renyi)


Let x(1)≤ x(2)≤ ...≤ x(r)≤ ...≤ x(n) be an ordered random sample of odd size (i.e., n=2m+ 1,m = 0,1,2, ...) from TWED(ω,λ ,β ). Therefore, the p.d.f. of X(1),X(n),X(r)

and X(m+1) are respectively given by (4.22), (4.23), (4.24) and (4.25).

fX1:n (x) = n

[1− γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

(Γ(ω +1))2

]n−1

×

λ e−λx (λx)ω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 . (4.22)

fXn:n (x) = n

[γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

(Γ(ω +1))2

]n−1

×

λe−λx (λx)ω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 . (4.23)

fXr:n (x) =nλe−λx (λx)ω

(r−1) !(n− r) !

[1− γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

(Γ(ω +1))2

]n−r

×

[γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

(Γ(ω +1))2

](r−1)[(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 .

(4.24)

fX(m+1):n (x) =(2m+1) !λe−λx (λx)ω

m!m!

[1− γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

[Γ(ω +1)]2

]m

×

[γ (ω +1,λx) [Γ(ω +1)+βΓ(ω +1,λx)]

[Γ(ω +1)]2

]m[(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 .

(4.25)


Theorem 4.4. The Bonferroni and Lorenz curve of TWED are respectively given by

B(p) =1

pµ

[2β

(Γ(ω +1))2

[Γ(2ω +2,qλ )

22ω+1 −Γ(ω +1,qλ )

2(qλ )ω+1

eqλ+(ω +1)Γ(ω +1,qλ )

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−2Γ(2ω +2)− (ω +1)4ω+1Γ(ω +1)2

22ω+3λ

]+

(1−β )γ(ω +2,λq)λΓ(ω +1)

]. (4.26)

L(p) =1µ

[2β

(Γ(ω +1))2

[Γ(2ω +2,qλ )

22ω+1 −Γ(ω +1,qλ )

2(qλ )ω+1

eqλ+(ω +1)Γ(ω +1,qλ )

−2Γ(2ω +2)− (ω +1)4ω+1Γ(ω +1)2

22ω+3λ

]+

(1−β )γ(ω +2,λq)λΓ(ω +1)

]. (4.27)

where, µ = µ′1, p ∈ [0,1], q = F−1(p).

Proof. From the definition of Bonferroni curve, we have

B(p) =1

pµ

∫ q

0x ft(x,ω,λ ,β )dx

B(p) =1

pµ′1

∫ q

0x

λ ω+1e−λxxω [(1−β )Γ(ω +1)+2βΓ(ω +1,λx)]

(Γ(ω +1))2 dx

B(p) =1

pµ

[(1−β )λ ω+1

Γ(ω +1)

∫ q

0e−λxxω+1dx+

2βλ ω+1

(Γ(ω +1))2

∫ q

0e−λxxω+1

Γ(ω +1,λx)dx

]

B(p) =1

pµ

[(1−β )γ(ω +2,λq)

λΓ(ω +1)+

2βλ ω+1

[Γ(ω +1)]2I3

], (4.28)

where, I3 is of the form:

∫ a

0e−bxxc

Γ(c,bx)dx =1bc

[Γ(2c,ab)

22c−1 −Γ(c,ab)

2(ab)ce−ab + cΓ(c,ab)

−2Γ(2c)− c4cΓ(c)2

22c+1b

].

Therefore, we can write

(4.29)I3 =

1λ ω+1

[Γ(2ω + 2,qλ )

22ω+1 −Γ(ω +1,qλ )

2(qλ )ω+1e−qλ +(ω +1)Γ(ω +1,qλ )

− 2Γ(2ω + 2)− (ω + 1)4ω+1Γ(ω + 1)2

22ω+3λ

]Using, (4.29) in (4.28), we obtained

B(p) =1

pµ

[2β

(Γ(ω +1))2

[Γ(2ω +2,qλ )

22ω+1 −Γ(ω +1,qλ )

2(qλ )ω+1

eqλ+(ω +1)Γ(ω +1,qλ )

−2Γ(2ω +2)− (ω +1)4ω+1Γ(ω +1)2

22ω+3λ

]+

(1−β )γ(ω +2,λq)λΓ(ω +1)

].

Hence, Lorenz curve is given by

L(p) =1µ

[2β

(Γ(ω +1))2

[Γ(2ω +2,qλ )

22ω+1 −Γ(ω +1,qλ )

2(qλ )ω+1

eqλ+(ω +1)Γ(ω +1,qλ )

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−2Γ(2ω +2)− (ω +1)4ω+1Γ(ω +1)2

22ω+3λ

]+

(1−β )γ(ω +2,λq)λΓ(ω +1)

].


Random numbers from TWED are generated by employing the inverse sampling methodwhich is discussed in Section 1.2.7.1. Thus, following the same procedure for the gen-eration of random numbers from TWED we wrote the following general algorithm inR programming language:

> rtwed<-function(n,w,b,L)

+ U=runif(n,0,1)

+ cdf<- function(x,w,b,L)

+ fn<-((gamma(w+1)-Igamma(w+1,L*x,lower=F))*(gamma(w+1)

+ +b*Igamma(w+1,L*x,lower=F)))/((gamma(w+1))^2)



+ fn<-function(x)cdf(x,w,b,L)-U[i]

+ uni<-uniroot(fn,c(0,10),lower=0,upper=1000)


+ return(data)


Let x1,x2, ...xn be a random sample of size n from TWED. Therefore, the log likelihoodfunction is given by.

log[l(Θ | x)] =n(ω +1) logλ −λ

n

∑i=1

xi +ω

n

∑i=1

logxi−2n log[Γ(ω +1)] (4.30)

+n

∑i=1

log[(1−β )Γ(ω +1)+2βΓ(ω +1,λxi)],where Θ = (ω,λ ,β ).

Differentiating (4.30) w.r.t. ω, λ and β and equating the derived gradients to zero weobtained the following system of equations:

n

∑i=1

(1−β )Γ(ω +1)Ψ(ω +1)+2β G3,02,3

(λx | 1, 1

0,0,ω+1

)+2βΓ(ω +1,λxi) log(λxi)

(1−β )Γ(ω +1)+2βΓ(ω +1,λxi)+

n logλ +n

∑i=1

logxi−2nΨ(ω +1) = 0. (4.31)

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n(ω +1)λ

−n

∑i=1

xi−n

∑i=1

2βλxω+1e−λx

(1−β )Γ(ω +1)+2βΓ(ω +1,λxi)= 0. (4.32)

n

∑i=1

2Γ(ω +1,λx)−Γ(ω +1)(1−β )Γ(ω +1)+2βΓ(ω +1,λxi)

= 0, (4.33)

where, Gm,np,q

(z | a1, ..., ap

b1, ..., bq

)is known as Meijer G function and is given by

Gm,np,q

(z | a1, ..., ap

b1, ..., bq

)=

12πi

∫L

∏nj=1 Γ(1−a j + s)∏

mj=1 Γ(b j− s)

∏pj=n+1 Γ(a j− s)∏

qj=m+1 Γ(1−b j + s)

zsds. (4.34)

The Meijer G function is related to the Gamma function by the following relation:

∂Γ(a+b,c)∂a

= G3,02,3

(c| 1, 1

0,0,a+b

)+Γ(a+b,c) log(c). (4.35)

It is not possible to obtain the M.L.E.’s of ω, λ and β in closed form by solvingsimultaneously (4.31), (4.32) and (4.33). Therefore, the function "optim" in R pro-gramming is used to obtain the numerical estimates of the underlined parameters byfitting TWED to some considered data sets.

4.9 Application

In this section, application of TWED in describing a real life and a simulated data setis shown. TWED is fitted to two types of data sets and comparison criteria viz., AIC,BIC and AICc are employed to investigate the distribution of best fit among its specialcases for the considered data sets. After the fitting of TWED and its special cases tothe data sets considered, estimates of M.L.E.’s and employed comparison criteria areestimated and reported in Table 4.3. Two types of data sets considered are given asfollows:


The real life data set considered is regarding remission times (in months) of 128 Blad-der cancer patients reported in Lee and Wang (2003) and is given as below:

00.08 02.09 03.48 04.87 06.94 08.66 13.11 23.63 00.20 02.23 03.52

04.98 06.97 09.02 13.29 00.40 02.26 03.57 05.06 07.09 09.22 13.80

25.74 00.50 02.46 03.64 05.09 07.26 09.47 14.24 25.82 00.51 02.54

03.70 05.17 07.28 09.74 14.76 26.31 00.81 02.62 03.82 05.32 07.32

10.06 14.77 32.15 02.64 03.88 05.32 07.39 10.34 14.83 34.26 00.90

02.69 04.18 05.34 07.59 10.66 15.96 36.66 01.05 02.69 04.23 05.41

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4.9. Application

07.62 10.75 16.62 43.01 01.19 02.75 04.26 05.41 07.63 17.12 46.12

01.26 02.83 04.33 07.66 11.25 17.14 79.05 01.35 02.87 05.62 07.87

11.64 17.36 01.40 03.02 04.34 05.71 07.93 11.79 18.10 01.46 04.40

05.85 08.26 11.98 19.13 01.76 03.25 04.50 06.25 08.37 12.02 02.02

03.31 04.51 06.54 08.53 12.03 20.28 02.02 03.36 06.76 12.07 21.73

02.07 03.36 06.93 08.65 12.63 22.69 05.49


Herein, a data set of size 100 is simulated from TWED with ω = 5, λ = 5 and β = 0.5by using the function "rtwed" defined above in Section 4.7 as:

>Simulateddata<-rtwed(100,5,0.5,5)

>head(Simulateddata)

[1] 0.7660 0.8714 1.0745 1.6769 0.6980 1.6408 1.8529 1.1779 1.1389

TABLE 4.3: M.L.E.’s and different comparison criteria.

Data Distn.M.L.E.’s

−2log l BIC AIC AICcω β λ

Rem

issi

onTi

me

TWED 0.270039 0.7111254 0.09362 822.282 833.139 828.28 828.476TED 0∗ -0.253742 0.12098 828.463 838.167 832.46 832.559

TLBED 1∗ 0.5811125 0.17429 843.465 853.169 847.46 847.561TABED 2∗ 0.529122 0.28140 908.769 918.473 912.77 912.865WED 0.172346 0∗ 0.12518 826.736 836.439 830.74 830.832ED 0∗ 0∗ 0.10677 828.684 833.536 830.68 830.716

LBED 1∗ 0∗ 0.21357 853.593 858.445 855.59 855.625ABED 2∗ 0∗ 0.32031 921.996 926.848 923.99 924.028

Sim

ulat

edda

ta

TWED 5.000380 0.499712 5.00070 82.4099 96.2255 88.409 88.6599TED 0∗ -1.000000 1.26465 155.304 164.515 159.31 159.429

TLBED 1∗ -0.999997 2.49925 106.107 115.318 110.11 110.232TABED 2∗ -0.999996 3.67172 87.0788 96.2892 91.079 91.2025WED 5.168760 0∗ 5.83644 87.4070 96.6174 91.407 91.5308ED 0∗ 0∗ 0.93582 213.266 217.872 215.27 215.307

LBED 1∗ 0∗ 1.72475 149.663 154.269 151.66 151.704ABED 2∗ 0∗ 2.71132 117.799 122.405 119.79 119.840

Observation with * as superscript refer to known quantities.

After the fitting of TWED to the underlined data sets, M.L.E.’s of parameters alongwith AIC, BIC and AICc are estimated for each of the special cases of TWED and aregiven in Table 4.3. From Table 4.3 it is observed that for the two considered data sets,it is TWED which is found to be the distribution of best fit because of possessing theleast AIC, AICc and BIC on fitting it to the underlined data sets.

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0 20 40 60 80

0.00

0.02

0.04

0.06

0.08

0.10

Remission times

D e

n s

i t y

EmpiricalTWEDTEDTLBEDTABEDWEDEDLBEDABED

(a) Remission Times

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

Remission times

Pr.(X

≤x)


(b) Remission Times

0.0 0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

Simulated data

D e

n s

i t y



(c) Simulated data

0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

Simulated data

Pr.(X

≤x)


(d) Simulated data


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4.10. Conclusion

4.10 Conclusion

In this Chapter, we have introduced transmuted weighted exponential distribution whichacts as a generalization to seven distributions, viz., TED, TLBED, TABED, WED,LBED, LBED and ED. After introducing TWED, we investigated its various structuralproperties. Two types of data sets are considered in order to make comparison betweenspecial cases of TWED in terms of distribution of best fit. The data sets considered areboth a real life and a simulated one. After the fitting of TWED and its special casesto the data sets considered it is found that TWED possesses least values of AIC, AICcand BIC on its fitting to both real life and simulated data set and hence proved to bethe distribution of best fit for the considered data sets. Distributions in order of best fitare give as follows:

(For Remission Times)(Best) TWED→ED→WED→TED→TLBED→LBED→TABED→ABED (Good)

(For Simulated data)(Best) TWED→TABED→WED→TLBED→ABED→LBED→TED→ED (Good)

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Chapter 5Weighted Gamma-Pareto Distribution

5.1 Introduction

As discussed in Chapter 1, there are various techniques that can be used to introducea new extension of an already existing probability model and results in a vast varietyof probability distributions. These techniques include the concept of G-distributions,mixture distribution, T-X family, truncation, transmutation and many more, e.g., twodifferent approaches of introducing the weighted version of an exponential distributionresult in the formation of two different models, viz., gamma and generalized exponen-tial distribution. Alzaatreh, Lee, and Famoye (2013) suggested a new technique toderive a vast family of probability distributions known as T-X family by consideringtwo different random variables one known as transformer and another the transformed.Since the concept of T-X family came into existence, different distributions of this fam-ily are introduced on considering various combinations of transformer and transformedrandom variables e.g., Alzaatreh, Famoye, and Lee (2013), Alzaatreha, Famoye, andLee (2014), Tahir et al. (2016) and Korakmaz (2018) etc.

This Chapter is devoted to the study of weighted version of gamma-Pareto distribu-tion (GPD). GPD was introduced by Alzaatreh, Famoye, and Lee (2012) on consider-ing gamma as transformed and Pareto as transformer random variables. The introducedweighted version of GPD is termed as weighted gamma-Pareto distribution (WGPD).Since, GPD is form-invariant under size biased sampling therefore a new weight func-tion which is different from w(x) = xω as has been used in the earlier chapters isassumed for the derivation of WGPD.

Interestingly, the introduced weighted version proved to be the generalization ofsome well known distributions, viz., baseline distribution, i.e., GPD by Alzaatreh,

? The content of this chapter is communicated for publication.

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Chapter 5. Weighted Gamma-Pareto Distribution

Famoye, and Lee (2012), weighted exponential-Pareto, exponential distribution, Paretodistribution, gamma distribution, weighted gamma distribution by Jain, Singla, andGupta (2014), weighted exponential by Gupta and Kundu (2009), and generalized ex-ponential distribution by Gupta and Kundu (1999).

5.1.1 Derivation of weighted gamma-Pareto Distribution

Alzaatreh, Famoye, and Lee (2012) introduced gamma-Pareto distribution after con-sidering T ∼ gamma(α,β ) as transformed and X ∼ Pareto(θ ,k) as a transformer ran-dom variable respectively. W (F(x)) =− log[1−Fα(x)]|α=1= H(x) given in Table 1.1of Chapter 1 were used for the derivation of GPD. The derived distribution and densityfunction of the resulting gamma-Pareto random variable are respectively given by

G(x;α,β ,k,θ) =γ(α,kβ−1 log(x/θ)

)Γ(α)

,x > θ ≥ 0,k,α,β > 0. (5.1)

g(x;α,β ,k,θ) =kα (θ/x)kβ−1

(log(x/θ))α−1

xβ αΓ(α),x > θ ≥ 0,k,α,β > 0. (5.2)

Re-parameterizing (5.1) and (5.2) by kβ−1 = λ we get

G(x;α,λ ,θ) =γ (α,λ log(x/θ))

Γ(α),x > θ ≥ 0,α,λ > 0. (5.3)

g(x;α,λ ,θ) =λ α (θ/x)λ (log(x/θ))α−1

xΓ(α),x > θ ≥ 0,α,λ > 0. (5.4)

where, γ (a,b) =∫ b

0 xa−1e−xdx is known as lower incomplete gamma integral.

Theorem 5.1. GPD is form invariant under size biased sampling.

Proof. The size biased version of order ω of a density d(x) is given by

dSB(x;ω) =xωd(x)∫xωd(x)dx

, ω > 0. (5.5)

Therefore, we can have the size biased version of GPD as:

gSB(x;α,λ ,θ ,ω) =xωg(x;α,λ ,θ)∫

∞

θxωg(x;α,λ ,θ)dx

gSB(x;α,λ ,θ ,ω) =

λ αθ λ xω−λ−1

Γ(α)(log(x/θ))α−1

∫∞

θ

λ αθ λ xω−λ−1

Γ(α)(log(x/θ))α−1 dx

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5.1. Introduction

gSB(x;α,λ ,θ ,ω) =xω−λ−1 (log(x/θ))α−1∫

∞

θxω−λ−1 (log(x/θ))α−1 dx

.

Put log(x/θ) = y in the denominator, we get

gSB(x;α,λ ,θ ,ω) =xω−λ−1 (log(x/θ))α−1

θ ω−λ∫

∞

0 e−(λ−ω)yyα−1dy

gSB(x;α,λ ,θ ,ω) =xω−λ−1 (log(x/θ))α−1

θ ω−λ Γ(α)(λ −ω)−α

gSB(x;α,λ ,θ ,ω) =(λ −ω)α (log(x/θ))α−1 (θ/x)λ−ω

xΓ(α). (5.6)

Which is again the density of GPD.

5.1.1.1 Weight function

Since, GPD is form invariant under size bias sampling as shown in Theorem 5.1, there-fore it is futile to study weighted version of GPD, using weight function xω . Thus, anew kind of weight function which is actually the c.d.f. of Pareto distribution withscale = θ and shape = λω is considered and is given by

w(x) = 1− (θ/x)λω ,θ ≥ 0,λ ,ω > 0. (5.7)

The reason to consider c.d.f. of Pareto distribution as a weight function is that it isa non-negative and w(x)−→1 as ω−→∞. This property of w(x) will be helpful to un-derstand some special cases of WGPD. Therefore, using the definition of weighteddistribution given by (1.52) in Section 1.3, we can write

gw(x;ω,α,λ ,θ) =w(x)g(x;α,λ ,θ)

E [w(x)], (5.8)

where E [w(x)] = 1−∫

∞

θ

(θ

x

)λωλ α (θ/x)λ (log(x/θ))α−1

xΓ(α)dx

E [w(x)] = 1−∫

∞

θ

(θ

x

)λ (ω+1)λ α (log(x/θ))α−1

xΓ(α)dx.

Put log(x/θ) = y, we get

E [w(x)] = 1− λ α

Γ(α)

∫∞

0e−λ (ω+1)yyα−1dy = 1− (ω +1)−α . (5.9)

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Therefore, on making use of (5.4), (5.7) and (5.9) in (5.8), we obtained the p.d.f. ofWGPD which is given by

gw(x;ω,α,λ ,θ) =λ αθ λ

(1− (θ/x)λω

)(log(x/θ))α−1

(1− (ω +1)−α)xλ+1Γ(α),x > θ ≥ 0,ω,α,λ > 0. (5.10)

Similarly, the c.d.f. and reliability function associated with WGPD are respectivelygiven by (5.11) and (5.12)

Gw(x;ω,α,λ ,θ) =γ (α,λ log(x/θ))− (ω +1)−αγ (α,λ (ω +1) log(x/θ))

Γ(α)(1− (1+ω)−α). (5.11)

Rw(x;ω,α,λ ,θ) = 1− γ (α,λ log(x/θ))− (ω +1)−αγ (α,λ (ω +1) log(x/θ))

Γ(α)(1− (1+ω)−α). (5.12)

Notation X ∼WGPD(ω,α,λ ,θ) wherever used in this chapter means a random vari-able X following WGPD. From Figure 5.1, it can be seen that WGPD is a positively

1.0 1.5 2.0 2.5 3.0

01

23

45

x

g ω(

x, ω

, α,

λ,

θ)

ω = 0.1, α = 1, λ = 5, θ = 1ω = 0.1, α = 1, λ = 10, θ = 1ω = 0.1, α = 5, λ = 5, θ = 1ω = 0.1, α = 5, λ = 10, θ = 1ω = 1, α = 1, λ = 5, θ = 1ω = 1, α = 1, λ = 10, θ = 1ω = 1, α = 5, λ = 5, θ = 1ω = 1, α = 5, λ = 10, θ = 1

(a) density

1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

G ω( x

, ω,

α,

λ, θ

)


(b) c.d.f.

1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

x

R ω( x

, ω,

α,

λ, θ

)


(c) reliability

2 4 6 8 10

02

46

8

x

h ω(

x, ω

, α,

λ,

θ)


(d) hazard rate

FIGURE 5.1: P.d.f., c.d.f., reliability and hazard rate curves at differentvalues of ω,α,λ and θ .

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5.2. Properties

skewed distribution with upside down bath tub hazard rate.

5.2 Properties

Theorem 5.2. If X ∼WGPD(ω,α,λ ,θ), then the transformation Y = log(X/θ) fol-lows weighted gamma distribution (WGD) introduced by Jain, Singla, and Gupta(2014) with scale parameter λ , shape parameters α and ω .

Proof. Using the Jacobian method, we can obtain the p.d.f. of Y as

gw(y) = gw(x = θey,ω,α,λ ,θ)J (5.13)

where, J =dxdy

= θey is the Jacobian of transformation y = log(x/θ) whose inverse is

x = θey.Therefore, on using (5.10) in (5.13), we get

gw(y) =λ αθ λ

[1− (θ/(θey))λω

](log(θey/θ))α−1

(1− (ω +1)−α)(θey)λ+1Γ(α)θey

gw(y) =λ α

(1− e−λωy

)yα−1e−λy

(1− (ω +1)−α)Γ(α),y > 0,ω,α,λ > 0.

which is same as the density of weighted gamma distribution introduced by Jain,Singla, and Gupta (2014).

Corollary 5.2.1. If X ∼WGPD(ω,α = 1,λ ,θ), then the transformation Y = log(X/θ)

follows:

i) weighted exponential distribution (WED) introduced by Gupta and Kundu (2009)with scale parameter λ and shape parameter ω .

ii) Jone’s modelΓ(a+b)Γ(a)Γ(b)

ce−acy(1−e−cx)b−1 with a =1ω, b = 2 and c = λω , see

Jones (2004) and Nadarajah and Kotz (2005).

Theorem 5.3. Density of WGPD can be expressed as a linear combination of densitiesof two gamma-Pareto distributions with respective scale parameters λ , λ (ω + 1) andwith same shape parameter α .

Proof. Equation (5.10) can be rewritten as

gw(x;ω,α,λ ,θ) = mλ αθ λ (log(x/θ))α−1

xλ+1Γ(α)−m

λ αθ λ (ω+1) (log(x/θ))α−1

xλ (ω+1)+1Γ(α)

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gw(x;ω,α,λ ,θ) = mλ αθ λ (log(x/θ))α−1

xλ+1Γ(α)− m

(ω +1)α

(λ (ω +1))αθ λ (ω+1) (log(x/θ))α−1

xλ (ω+1)+1Γ(α)

gw(x;ω,α,λ ,θ) = m g(x;α,λ ,θ)− m(ω +1)α

g(x;α,λ (ω +1),θ),

where, m = (1− (ω +1)−α)−1.

Theorem 5.4. If X ∼ GPD(α,λ ,θ) and Xw ∼ WGPD(ω,α,λ ,θ) then X is smallerthan Xw in stochastic, failure rate, likelihood ratio and mean residual life ordering.

Proof. To prove the statement, it is enough to show that X ≤LR Xw as it implies theother three partial orderings due to chain of implications given by (1.40) in Section1.2.9.Dividing (5.4) by (5.10) we get the following ratio:

g(x;α,λ ,θ)

gw(x;ω,α,λ ,θ)=

1− (ω +1)−α

1− (θ/x)λω. (5.14)

Differentiating (5.14) w.r.t. x we get

∂

∂xg(x;α,λ ,θ)

gw(x;ω,α,λ ,θ)=−λω [1− (ω +1)−α ]θ λω

xλω+1[1− (θ/x)λω

]2 < 0 ∀x,ω,α,λ ,θ . (5.15)

Since, the gradient of ratio of two densities w.r.t. x comes out to be negative im-plying that it is decreasing in x. Hence, it can be concluded that X ≤LR Xw and the restof three partial orderings follow according, i.e., X ≤FR Xw,X ≤ST Xw and X ≤MR Xw.X ≤LR Xw can also be justified by using the statement of Theorem 1.1 (Patil, Rao, andRatnaparkhi (1986)) which states that if the considered weight function is monotonicincreasing then the corresponding weighted version of a random variable is greaterthan the baseline random variable in terms of likelihood ratio ordering. Since theweight function considered here, is the c.d.f. of Pareto distribution which is monotonicincreasing. Thus, it can be concluded that WGPD is greater that GPD in likelihoodand hence due to chain of implication 1.40 it is greater than GPD also in terms ofstochastic, failure rate and mean residual life ordering.

Theorem 5.5. The rth moment about origin of WGPD is given by

µ′r =

λ αθ r ((λ − r)−α − (λω +λ − r)−α)

(1− (ω +1)−α),λ > r,r = 1,2,3, ... (5.16)

Proof. From the definition of rth moment about origin, we can write

µ′r =

∫∞

θ

xrgw(x;ω,α,λ ,θ)dx

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5.2. Properties

µ′r =

∫∞

θ

xrλ αθ λ

(1− (θ/x)λω

)(log(x/θ))α−1

(1− (ω +1)−α)xλ+1Γ(α)dx

µ′r =

λ αθ λ

(1− (ω +1)−α)Γ(α)

∫∞

θ

xr−λ−1(

1− (θ/x)λω)(log(x/θ))α−1 dx.

On using the substitution log(x/θ) = y, we get

µ′r =

λ αθ r

(1− (ω +1)−α)Γ(α)

∫∞

0e(r−λ )y

(1− e−λωy

)yα−1dy

µ′r =

λ αθ r

(1− (ω +1)−α)Γ(α)

(∫∞

0yα−1e−(λ−r)ydy−

∫∞

0yα−1e−(λω+λ−r)ydy

)µ′r =

λ αθ r

(1− (ω +1)−α)Γ(α)

(Γ(α)

(λ − r)α− Γ(α)

(λω +λ − r)α

)µ′r =

λ αθ r ((λ − r)−α − (λω +λ − r)−α)

(1− (ω +1)−α),λ > r.

First four moments about origin are obtained respectively by substituting r = 1,2,3,4 in (5.16) and are given as

µ′1 =

λ αθ ((λ −1)−α − (λω +λ −1)−α)

(1− (ω +1)−α),λ > 1. (5.17)

µ′2 =

λ αθ 2 ((λ −2)−α − (λω +λ −2)−α)

(1− (ω +1)−α),λ > 2. (5.18)

µ′3 =

λ αθ 3 ((λ −3)−α − (λω +λ −3)−α)

(1− (ω +1)−α),λ > 3. (5.19)

µ′4 =

λ αθ 4 ((λ −4)−α − (λω +λ −4)−α)

(1− (ω +1)−α),λ > 4. (5.20)

Similarly, variance is given by

σ2 =

θ 2λ α

((ω +1)−α −1)2

[1− (ω +1)−α

(λ −2)−α − (λω +λ −2)−α

−λ

α(λ −1)−α − (λω +λ −1)−α

2],λ > 2. (5.21)

Mathematical expression for the coefficient of variation, skewness and kurtosis arenot given in closed form but their estimates are obtained numerically at different valuesof parameters and are reported in Table 5.1. From Table 5.1 it is can be observed that:

• on increasing θ , mean and variance increases whereas coefficient of variation,skewness and kurtosis remains unaltered.

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TABLE 5.1: Characteristic of WGPD at different values of ω,α,λ andθ .

θ λ α ωMean Variance CV Skewness Kurtosis

µ σ2 cv γ1 γ2

5

55

0.5 16.022 95.678 0.611 5.452 183.2271.0 15.478 89.848 0.612 5.515 187.587

100.5 47.019 1986.930 0.948 9.355 1522.9121.0 46.598 1967.018 0.952 9.366 1528.816

155

0.5 7.179 1.315 0.160 1.539 4.5731.0 7.096 1.283 0.160 1.564 4.679

100.5 10.005 5.204 0.228 1.503 4.5201.0 9.971 5.211 0.229 1.499 4.496

10

55

0.5 32.044 382.710 0.611 5.452 183.2271.0 30.956 359.391 0.612 5.515 187.587

100.5 94.038 7947.719 0.948 9.355 1522.9121.0 93.195 7868.071 0.952 9.366 1528.816

155

0.5 14.357 5.259 0.160 1.539 4.5731.0 14.193 5.132 0.160 1.564 4.679

100.5 20.009 20.816 0.228 1.503 4.5201.0 19.942 20.842 0.229 1.499 4.496

• increase in λ leads to decrease in all of the five measures, i.e., mean, variance,coefficient of variation, skewness and kurtosis.

• with the increase in α , mean, variance and coefficient of variation increaseswhereas skewness and kurtosis shows unevenness, i.e., it initially increases andthen starts decreasing with the increase in λ .

• increase in ω leads to decline in mean and variance whereas the other threemeasures, viz., coefficient of variation, skewness and kurtosis increases.

5.3 Entropy measure

Theorem 5.6. Renyi entropy associated with WGPD is given by

HR(δ ) =1

1−δlog

[λ δαΓ(δ (α−1)+1)

(1− (ω +1)−α)δ (Γ(α))δ θ δ−1

∞

∑k=0

(−1)δ+k(

δk

)(λωδ +λδ +δ −λωk−1)δ (α−1)+1

],

δ ≥ 0,δ 6= 1. (5.22)

Proof. From the definition of Renyi entropy given by (1.34) in Section 1.2.6.1, we canwrite

HR(δ ) =1

1−δlog

[∫∞

θ

λ δαθ δλ [1− (θ/x)λω ]δ [log(x/θ)]δ (α−1)

[1− (ω +1)−α ]δ xδ (λ+1)(Γ(α))δdx

]

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Substitute log(x/θ) = y we get

HR(δ ) =1

1−δlog

[λ δαθ δλ [1− (ω +1)−α ]

−δ

(−1)−δ [Γ(α)]δ θ δ (λ+1)−1

∫∞

0(e−λωy−1)δ e−(δ (λ+1)−1)yyδ (α−1)dy

]

Put λωy = t we get

HR(δ ) =1

1−δlog

[λ δαθ δλ [1− (ω +1)−α ]

−δ

(−1)−δ [Γ(α)]δ θ δ (λ+1)−1 ×

∫∞

0(e−t −1)δ exp

(−(δ (λ +1)−1)

λωt)

tδ (α−1)

(λω)δ (α−1)+1 dt

]. (5.23)

Consider the following integral:

∫∞

0(e−t−1)ne−mttν−1dt = Γ(ν)

∞

∑k=0

(−1)k (nk

)(n+m− k)ν

. (5.24)

Therefore, by making use of (5.24) with n = δ ,m =(δ (λ +1)−1)

λω,ν = δ (α−1)+1

in (5.23) we get

HR(δ ) =1

1−δlog

[λ δαΓ(δ (α−1)+1)

(1− (ω +1)−α)δ (Γ(α))δ θ δ−1

∞

∑k=0

(−1)δ+k(

δk

)(λωδ +λδ +δ −λωk−1)δ (α−1)+1

].


Random numbers from WGPD are generated by employing Rejection method whichis discussed in Section 1.2.7.2, on treating the Pareto with shape=k and scale=θ1 asproposal density. Thus, the ratio of target to proposal density is given by

ρ(y) =gw(y;ω,α,λ ,θ)

kθ1y−(k+1)

ρ(y) =λ αθ λ

(1− (θ/y)λω

)(log(y/θ))α−1 y(k−λ )

(1− (ω +1)−α)Γ(α)kθ1. (5.25)

Differentiating (5.25) w.r.t. y as:

∂ρ(y)∂y

=1

kθ1 ((ω +1)α −1)Γ(α)

[xk−1

(λ (ω +1))−α

(θ

x

)λ (1+ω)(log( x

θ

))α−2×

log( x

θ

)((k−λ )

( xθ

)λω

+λω− k+λ

)+(α−1)

(( xθ

)λω

−1)]

. (5.26)

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To determine the constant m such that ρ(y)≤ m ∀ y, we solve the following equationfor y at preassigned parameter values of target and proposal density:

∂ρ(y)∂y

= 0 (5.27)

After obtaining the critical value m on solving (5.27) such that∂ 2ρ(y)

∂y2 |y=m< 0, the

following two steps are followed:

i) Simulate random numbers Y = y and U = u respectively from the Pareto(θ1,k)

density and U(0,1) density.

ii) Retain Y = y and consider it a random number from WGPD(ω,α,λ ,θ) if u ≤ρ(y)

motherwise return to step (i).

The above discussed procedure/algorithm is written in R programming as:

> VNsample<-function(n,n1,xstart,w,a,L,t,scale,shape)#w=omega,

+ #a=alpha,L=lambda,t=theta,scale and shape of proposal density,

+ #n=required size, n1=size generated from proposal density

+ Ratio<-function(x)eval(w;a;L;t;scale;shape;((Lâ)/(gamma(a)*x*

+ (1-(1+w)^(-a))))*(1-(t/x)^(L*w))*((t/x)^L)*((log(x/t))^(a-1))/(

+ scale*x^(-(shape+1))))

+ D1<-function(x)eval(w;a;L;t;scale;shape;DD(expression(((Lâ)/(

+ gamma(a)*x*(1-(1+w)^(-a))))*(1-(t/x)^(L*w))*((t/x)^L)*((log(x/t

+ ))^(a-1))/(scale*x^(-(shape+1)))),"x",1))

+ D2<-function(x)eval(w;a;L;t;scale;shape;DD(expression(((Lâ)/(

+ gamma(a)*x*(1-(1+w)^(-a))))*(1-(t/x)^(L*w))*((t/x)^L)*((log(x/t)

+ )^(a-1))/(scale*x^(-(shape+1)))),"x",2))

+ library(nleqslv)

+ Criticalpoint=nleqslv(xstart, D1)#xstart=guess value

+ d2=D2(Criticalpoint$x)# second deriavtive test

+ if(d2 >= 0) stop("The point is not the point of Maximum")

+ if(Criticalpoint$x<t)stop("Abscissa is less than Min(Support)")

+ m=Ratio(Criticalpoint$x)#Ordinate of Ratio at critical point

+ curve(Ratio(x),xlim=c(t,t+150),xlab=expression(y),

+ ylab=expression(rho(y)),bty="l")

+ abline(v=Criticalpoint$x,h=Ratio(Criticalpoint$x),lty=2)

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5.5. Maximum likelihood estimation

+ points(Criticalpoint$x,Ratio(Criticalpoint$x),cex=1,pch=19,col=2)

+ library(VGAM)

+ y<-rpareto(n1,scale,shape)

+ u<-runif(n1)

+ data=c()

+ for(i in 1:n1)

+ if (u[i]<=(1/m)*Ratio(y[i]))

+ data=c(data,y[i])

+ if (length(data)==n)break

+ L=list(Data=data,Point_Max.=c(Criticalpoint$x,

+ Ratio(Criticalpoint$x)))

+ return(L)


Let θ < x(1) ≤ x(2) ≤ ...≤ x(n) be an ordered sample of size n from WGPD. Thereforeits log likelihood function is given by

log[l(Θ|x)] = nα logα−n

∑i=1

logxi−n log(Γ(α))−n log[1− (ω +1)−α

]+nλ logθ

+(α−1)n

∑i=1

log(

log(xi

θ

))−λ

n

∑i=1

logxi +n

∑i=1

log

(1−(

θ

xi

)λω). (5.28)

Before moving forward to find the estimates of parameters, it is worth to discuss theprocedure of estimating an unknown parameter, support of the distribution depends on.Smith (1985) discussed the maximum likelihood estimation in a class of non-regularcases wherein the probability density is zero for x less than an unknown parameter θ .Distribution of such nature considered by Smith (1985) are three-parameter versionsof Weibull, gamma, beta and log gamma. The density of all these distributions is zerofor x less than an unknown parameter θ in other words one can say that support of thedistribution depends on θ . Smith (1985) suggested to estimate the unknown parame-ter θ by sample minimum x(1) and the remaining parameters by maximum likelihoodestimation after excluding x(1) from the sample. In parameter estimation of WGPDthe case is same that the density is zero for x < θ moreover the likelihood functionis increasing w.r.t. θ . Therefore the M.L.E. of θ under the restriction x < θ is x(1),i.e., θmle = x(1) The M.L.E.’s of remaining three parameters, viz., α,ω and λ are ob-tained by using maximum likelihood estimation procedure after excluding x(1) fromthe sample.

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Differentiating the log likelihood function w.r.t. α,ω,λ after replacing θ by itsM.L.E. x(1) and excluding x(1) form the sample. Equating respective gradients of loglikelihood to zero, we get the following system of three nonlinear equations:

n logλ −nΨ(α)+n log(1+ω)

1− (1+ω)α+ ∑

xi 6=x(1)

log

(log

(xi

x(1)

))= 0. (5.29)

∑xi 6=x(1)

λ log(x(1)/xi

)(x(1)/xi

)λω(x(1)/xi

)λω −1− nα

(1+ω)((1+ω)α −1)= 0. (5.30)

nα

λ+n logx(1)− ∑

xi 6=x(1)

ω(x(1)/xi)λω log

(x(1)/xi

)1− (x(1)/xi)λω

− ∑xi 6=x(1)

logxi = 0. (5.31)

It is impossible to solve the above system of equations manually because of beingnon linear in nature. Therefore the function "nleqslv" is used to solve it in R pro-gramming. The above mentioned system of equations is defined in R programming asfollows:

> system<- function(x,D) # x=vector of unknowns

+ t=min(D) # t=theta, D= a numeric vector, i.e., data set

+ d<-D[D!=min(D)] # exclude sample minimum

+ n<-length(d)

+ y <- numeric(3) # no. of equations

+ #Grandient equation w.r.t. omega (x[1])

+ y[1]=sum(sapply(1:n,function(i)((t/d[i])^(x[3]*x[1])*x[3]*

+ log(t/d[i]))/((t/d[i])^(x[3]*x[1])-1)-x[2]/((1+x[1])*

+ ((1+x[1])^x[2]-1))))

+ #Grandient equation w.r.t. aplha (x[2])

+ y[2]=sum(sapply(1:n,function(i)x[2]/x[3]+log(t)-((t/d[i])^(x[3]*

+ x[1])*x[1]*log(t/d[i]))/(1-(t/d[i])^(x[3]*x[1]))-log(d[i])))

+ #Grandient equation w.r.t. lambda (x[3])

+ y[3]=sum(sapply(1:n,function(i)log(x[3])-digamma(x[2])+(log(1+

+ x[1]))/(1-(1+x[1])^(x[2]))+log(log(d[i]/t))))

+ y

The above defined system of equations can be solved by using function "nleqslv"

with the following general syntax in R programming:

>nleqslv(xstart, S, data,...)

xstar= vector of guess values.

S= name of the system to get solved.

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5.6. Application

data= numeric vector to be supplied.

...= additional arguments.

5.6 Application

For illustration purpose two types of data sets are considered and WGPD along with itsspecial cases is fitted to them. Performance of WGPD and its special cases is studiedin terms of fitting and the model of best fit for the considered data sets is found byusing a model selection tools, viz., AIC. Two types of data sets considered include areal life and a simulated one, and are given as follows:


A random sample of size 100 is simulated from WGPD with ω = 0.5,α = 5,λ =

5,θ = 10 by employing the algorithm given above in Section 5.4. The values of scaleand shape of the proposal density is taken 10 and 1 respectively. Therefore the requiredsample can be generated by executing the following R command:

> Simulated_data<-VNsample(100,500,20,0.5,5,5,10,10,1)

> Simulated_data

$Data

[1] 22.85 142.12 18.95 21.17 25.27 30.64 33.98 24.13 24.20 23.22

[11] 36.09 36.14 73.39 47.21 26.88 84.48 27.67 38.23 37.15 33.69

[21] 17.77 28.97 18.52 24.50 20.10 49.70 32.45 18.72 15.36 19.35

[31] 15.20 64.05 33.93 14.56 24.95 25.20 37.70 55.20 28.85 46.61

[41] 15.25 45.58 25.38 27.31 48.15 22.67 60.37 28.69 22.18 16.65

[51] 99.43 17.21 59.21 19.28 54.98 24.19 14.07 22.82 21.01 26.34

[61] 22.52 36.16 16.99 27.26 19.63 37.50 29.20 35.24 17.69 65.48

[71] 27.19 45.12 15.41 16.79 19.06 22.63 24.88 27.36 17.46 62.47

[81] 20.03 72.01 20.81 29.99 18.89 15.84 37.78 46.55 73.07 31.18

[91] 21.82 45.09 21.72 30.30 28.53 47.77 34.53 33.35 24.11 45.24

$Point_Max.

[1] 28.609356 2.535282

From Figure 5.2 it is clear that the ratio of target to proposal density (ρ(y)) at assignedvalues of parameters used for generation of random numbers, is increasing functionand the value of constant m comes out to be 2.535282.

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20 40 60 80 100 120

0.00.5

1.01.5

2.02.5

y

ρ(y)

m = 2.535282

y = 28.60936

FIGURE 5.2: Ratio of target to proposal density at ω = 0.5,α = 5,λ =5,θ = 10,θ1 = 10,k = 1 .


The real life data set considered is related to flood discharge which is reported in Alza-atreh, Famoye, and Lee (2012) and is given as follows:

1460 4050 3570 2060 1300 1390 1720 6280 1360 7440 5320 1400 3240

2710 4520 4840 8320 13900 71500 6250 2260 318 1330 970 1920 15100

2870 20600 3810 726 7500 7170 2000 829 17300 4740 13400 2940 5660

After the fitting of WGPD and its special cases to the considered data sets, M.L.E.’salong with AIC are estimated, and are given in Table 5.2. From Table 5.2 it can be no-ticed that for simulated data set WGPD possesses least AIC followed by GPD, WGD,GD, PD, WED, GED, ED and WEPD respectively. Whereas, for the real life data setit is GPD which proves to be the distribution of best fit followed in order by WGPD,ED, GD, WED, WGD, GED, PD and GEPD.

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5.6. Application

TABLE 5.2: M.L.E.’s, -log likelihood and AIC

DataDist

n. M.L.E.’s-log(l) AIC

θmle αmle λmle ωmle

Sim

ulat

edD

ata

WGPD 14.05 1.8981 3.0544 4.7707 290.5741 589.1482WEPD 14.05 ... 0.1078 119.0706 559.5579 1125.116GPD 14.05 2.1436 3.3464 ... 366.4171 738.8342WGD ... 8.2555 0.2914 0.756900 370.4862 746.9724WED ... ... 0.0703 0.000002 403.3208 810.6416GED ... ... 0.0514 ... 405.9222 813.8445GD ... 7.1697 0.2520 ... 373.4016 750.8031ED ... ... 0.0351 ... 434.8147 871.6294PD 14.05 ... 1.5769 ... 382.1350 768.2700

Floo

ddi

scha

rge

WGPD 318 6.063 2.441327 1.6153 365.4407 738.8813WEPD 318 ... 0.042914 40.506 437.8323 881.6646GPD 318 6.1351 2.4657 ... 365.4521 736.9042WGD ... 1.7995 0.000304 2.6476 385.9541 777.9082WED ... ... 0.0002 17.44 383.4422 770.8843GED ... ... 0.000234 ... 388.9989 779.9978GD ... 0.9196 0.000136 ... 382.9048 769.8097ED ... ... 0.000148 ... 382.9964 767.9929PD 318 ... 0.412714 ... 392.8100 789.6200

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20 40 60 80 100 120 140

0.00

0.02

0.04

0.06

0.08

Simulate Data

Den

sity

EmpiricalWGPDWEPDGPDWGDWEDGDEDPDGED

(a) Simulated data

50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

1.0

Simulate Data

Pr.(

X≤

x)EmpiricalWGPDWEPDGPDWGDWEDGDEDPDGED

(b) Simulated data

0 10000 20000 30000 40000 50000 60000

0.00

000

0.00

004

0.00

008

0.00

012

Flood_Discharge

Den

sity


(c) Flood Discharge

0 5000 10000 15000 20000 25000 30000 35000

0.0

0.2

0.4

0.6

0.8

1.0

Flood discharge

Pr.(

X≤

x)


(d) Flood Discharge


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5.6. Application

TAB

LE

5.3:

Spec

ialc

ases

ofW

GPD

Dis

trib

utio

nL

imiti

ngca

sean

dTr

ansf

orm

atio

nPr

obab

ility

dens

ityfu

nctio

n(p

.d.f.

)

GPD

(gam

ma-

Pare

todi

stri

butio

nin

trod

uced

byA

lzaa

treh

,Fam

oye,

and

Lee

(201

2))

ω→

∞λ

α

xΓ(α

)

[ log( x θ

)] α−1( θ x

) λ ,x>

θ.

(or)

ω→

0λ

α+

1

xΓ(α

+1)

[ log( x θ

)] α(θ x

) λ ,x>

θ.

WE

PD(W

eigh

ted

expo

nent

ial-

Pare

todi

stri

butio

n)α=

1[ 1−(ω

+1)−

1] −1[ 1−( θ x

) λω] θ

λ

xλ+

1,x

>θ

.

PD(P

aret

odi

stri

butio

n)α=

1,ω→

∞λ

θλ

xλ+

1,x

>θ

.

WG

D(W

eigh

ted

gam

ma

dist

ribu

tion

intr

oduc

edby

Jain

,Sin

gla,

and

Gup

ta(2

014)

)y=

log( x θ

)[1−(ω

+1)−

α]−

1[ 1−

e−λ

ωy] λα

e−λ

y

Γ(α

)yα−

1 ,y>

0.

GD

(Gam

ma

dist

ribu

tion)

y=

log( x θ

) ,ω→

∞λ

αe−

λy

Γ(α

)yα−

1 ,y>

0.

(or)

y=

log( x θ

) ,ω→

0λ

α+

1 e−λ

y

Γ(α

+1)

yα,y

>0.

WE

D(W

eigh

ted

expo

nent

iald

istr

ibut

ion

intr

oduc

edby

Gup

taan

dK

undu

(200

9))

y=

log( x θ

) ,α=

1[ 1−(ω

+1)−

1] −1[ 1−

e−λ

ωy] λ

e−λ

y ,y>

0.

GE

D(G

ener

aliz

edE

xpon

entia

ldis

trib

utio

n.(w

ithlo

catio

n=

0,sh

ape

=2,

y=

log( x θ

) ,α=

1,ω=

12λ[ 1−

e−λ

x] e−λ

x ,y>

0.

scal

e=

1/λ

)(se

eGup

taan

dK

undu

(199

9)))

ED

(Exp

onen

tiald

istr

ibut

ion)

y=

log( x θ

) ,ω→

∞,

α=

1λ

e−λ

y ,y>

0.

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5.7 Conclusion

In this Chapter, weighted version of gamma Pareto distribution is studied. It has beenshown that gamma-Pareto distribution is form invariant under size biased sampling.Therefore, a new weight function which is the c.d.f. of Pareto distribution is suggestedfor the first time. The weight function considered in this chapter helped to gener-alize various known distributions, viz., gamma, exponential, Pareto, gamma-Pareto,generalized exponential, weighted gamma, weighted exponential-Pareto and weightedexponential distribution. It has been shown that WGPD can be expressed as linearcombination of gamma-Pareto distributions with different scale and same shape pa-rameters. It is also shown that weighted gamma-Pareto distribution is greater than thebase line distribution, i.e., GPD in terms of stochastic, failure rate, likelihood ratio andmean residual life ordering. The random numbers from WGPD are generated by em-ploying the Rejection method after considering the Pareto as proposal density. Twotypes of data sets are considered including a real life and a simulated one. The sim-ulated data set is generated at ω = 0.5,α = 5,λ = 5,θ = 10 with shape and scale ofproposal density as θ1 = 10,k = 1 respectively. Real life data set considered is relatedto flood discharge. WGPD along with its special cases is fitted to the considered datasets and by using AIC as a model selection tool, it has been shown WGPD proves to bethe model of best fit for the simulated data set followed respectively by GPD, WGD,GD, PD, WED, GED, ED and WEPD. Whereas, for the considered real life data set itis GPD which proved to be the model of best fit followed by WGPD, ED, G, WED,WGD, GED, PD and GEPD respectively.

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