International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

On the Construction of Kumaraswamy-Epsilon

Distribution with Applications

Gongsin Isaac Esbond1 Saporu W O Funmilayo

2

1Department of Mathematical Sciences University of Maiduguri PMB 1069 Bama Road Maiduguri Borno State Nigeria

2National Mathematical Centre Abuja-Lokoja Expressway Kwali-Sheda P M B 118 Abuja Nigeria

Abstract A new probability distribution function of the Kumaraswamy-G family is introduced in this study It is constructed from a

parent distribution called the epsilon distributionThis new distribution has more flexible probability density function shapes that suggest

its possible use for example in modeling lifetime data generation processes Its hazard rate functions and order

statisticsdistributionswere derived The distribution significantly provided good fit to two real life datasets with efficient parameter

estimates within twice their standard errors These attest to its practical applicability

Keywords Kumaraswamy-G epsilon distribution hazard rate function order statistic deuterium excesses monthly tax revenue

1 Introduction

A beta-type distribution was introduced in 1980 for a double

bounded random variable [16] andnamed after the founder

Kumaraswamy A random variable 119883 is said to be

distributed according to the Kumaraswamy distribution with

parameters 119886 and 119887 if its density and cumulative distribution

functions are given respectively by

where in each case 0 lt 119909 lt 1 and119886 119887 gt 0The

distribution has application in hydrology reliability studies

[17] and other areas

Following the works of Jones [14] and Eugene et al [12]

Cordeiro and de Castro [6] proposed the use of

Kumaraswamy distribution as a generator of probability

distributions of continuous random variables with parent

distribution 119866 119909 These are referred to as the

Kumaraswamy-G distribution Its cumulative distribution

function is given by

where 119892 119909 =119889

119889119909119866 119909 The parameters 119886 and 119887 in the

equation (4) determine the skewness and tail weight

respectively

In this study a new distribution based on the Kumaraswamy

generator is introducedIt is called the Kumaraswamy-

epsilon distribution and subsequently denoted by the K-

epsilon distribution

2 Literature Review

Of recent interest in deriving new generalized probability

distributions has grown tremendously It has resulted in

generalized classes of univariate distributions after

introducing additional shape parameter to the baseline or

parent distributions The new generated families extend the

well-known classical probability distributions and at the

same time provide great flexibility in modeling data

generating processes For examples beta generator [12]

gamma generator [19] the Weibull-G family [3]

exponentiated family and generalized exponentiated family

[8] of density functions have all been introduced Many

probability distributions have been constructed using these

generators

Many members of the Kumaraswamy generator have been

constructed and applied For instance the Kumaraswamy

Gumbel [7] Kumaraswamy Laplace [1] Kumaraswamy

Pareto [4] Kumaraswamy generalized Rayleigh [13] the

Kumaraswamy Lindley distribution [5] Kumaraswamy

generalized gamma distribution [10] have been used in

modeling bathtub shaped hazard rate functions Also they

have the ability to model monotone and non-monotone

failure rate functions as such it can be useful in both

lifetime data analysis and reliability studies Oguntunde et al

[18] introduced the Kumaraswamy inverse exponential

distribution and studied some of its properties The

Kumaraswamy Weibull distribution was introduced in [9]

and used in the study of failure data

3 The K-epsilon distribution and its properties

The epsilon distribution was introduced in [11] and shown to

be asymptotically equivalent to the exponential distribution

For a random variable 119883 distributed according to the epsilon

distribution its probability density and cumulative

distribution functions are given respectively by

where 0 lt 119909 lt 120575 120575 120582 gt 0

Putting (6) into (3) and (4) the cumulative distribution and

probability density functions of the K-epsilon distribution

are given respectively by

Paper ID ART20202623 1021275ART20202623 1199

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

whereΥ = 1205752

1205752minus1199092 Ζ = 1 minus 119909+120575

120575minus119909 minus120582

2120575

0 lt 119909 lt 120575119886 119887 120575 120582 gt

0

The skewness and tail weight of the distribution are

controlled by the parameters 119886 and 119887 The plots of the K-

epsilon density function (8) at various parameter values are

given in Fig 1 below

Figure 1 K-Epsilon probability density plots at various parameter values

From (7) the quantile function of the K-epsilon distribution

is given by

where 0 lt 119901 lt 1 119886 119887 120575 120582 gt 0

Proposition 1 The K-epsilon distribution is a true probability density

function with unimodal shape property That is we want to

show that

(i) 119891119883 119909 is unimodal and

(ii) 119891119883 119909 120575

0119889119909 = 1

The proof of this proposition is given in the appendix

31 Moments of the K-epsilon distribution

The 119903119905119893 119903 = 1 2hellip moment of the K-epsilon can be

derived from the expression

From (10) it is obvious that the moments of the K-epsilon

distribution cannot be obtained in explicit form but can be

obtained by numerical integration for the desired moment(s)

when the parameter values are specified

32 Survival and hazard rate functions

Survival and hazard rate functions form the focal points in

survival analysis and reliability theory The survival

function which is the complementary cumulative

distribution or reliability function is defined as the

probability of surviving to the age of 119909 or exceeding age 119909

On the other hand the hazard rate function also known as

instantaneous failure rate or force of mortality is interpreted

as the amount of risk of failure associated with a unit of age

119909 The two functions are related however the hazard rate

a = 2 b = 04 120575 120582 = 4 (black) 25 (red)

15 (blue) 05 (green)

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

a = 2 120582 = 3 120575 = 6

b = 015 (black) 025 (red)

035 (blue) 05 (green)

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

a = 2 b = 04 120582 120575 = 6 (black) 10 (red)

15 (blue) 25 (green)

0 1 2 3 4 5 6X

values

00

02

04

06

08

Den

sity

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

b = 04 120582 = 3 120575 = 6

a = 2 (black) 3 (red)

5 (blue) 10 (green)

Paper ID ART20202623 1021275ART20202623 1200

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

function is more informative about the underlying

mechanism of failure

The survival function of the K-epsilon distribution is given

by

and its hazard rate function is given by

The plots of the hazard rate function of the K-epsilon

distribution at various parameter values are presented in Fig

2 below The shapes suggest that the K-epsilon hazard rate

function could be a good choice model for somefailure rate

datasets

Figure 2 K-Epsilon distribution hazard rate function at various parameter values

33 Order statistics distribution

Order statistics play important role in life testing and

reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of

a random variable 119883 having the K-epsilon distribution and

density functions 119865119883 and 119891119883 respectively The distribution of

the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip

119909(119903) hellip 119909(119899) is given by

119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)

Where Λ =119899

119903minus1 119899minus119903 Υ =

1205752

1205752minus1199092

Ζ = 1 minus 119909+120575

120575minus119909 minus120582

2120575

andΨ = 1minus Ζ119886 From (13) the K-epsilon

distribution of the first order statistic 119883(1) and the 119899119905119893 order

statistic 119883(119899) are given respectively by

4 Parameter Estimation

As noted earlier the explicit expressions for the K-epsilon

probability density cumulative distribution and quantile

functions has made the distribution a good candidate for the

deployment of the fitdistrplus package in R This has the

advantage of producing parameter estimates their standard

errors and many other important statistical properties

5 Application

The K-epsilon distribution is applied to 2 datasets

namelydeuterium excesses and tax revenue datasets These

are represented in boxplots in Fig 3 below

Paper ID ART20202623 1021275ART20202623 1201

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 3 Boxplots of Deuterium excesses and Monthly Tax

Revenue

The first dataset is hourly record of deuterium (heavy

hydrogen ndash one of the two stable isotopes of hydrogen)

excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]

given in the appendix

Parameter estimates and Kolmogorov-Smirnov goodness-of-

fit test results are given in Table 1 while the fitted K-epsilon

densityplot is presented in Fig 3

Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

From Table 1 it could be observed that the K-epsilon

distribution has fitted the deuterium excesses datasetThe

results of the goodness-of-fit test shown in Table 1 above

provide sufficient evidence that the K-epsilon distribution

fits the dataset Also the standard errors of the parameter

estimates obtained indicate they are of good precision

Figure 3 K-Epsilon distribution fit of hydrogen

Isotope (Deuterium) excesses

The second dataset for application is the monthly records of

tax revenue of Egypt between January 2006 and November

2010 in 1000 million Egyptian pounds reported in [15] It is

presented in the appendix

The results for parameter estimate and goodness-of-fit test

are presented in Table 2 while the fitted K-epsilon

probability density plot is presented in Fig 4

Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

Table 2 also suggest that the K-epsilon distribution has

successfully fitted the tax revenue dataset The parameter

estimates are of good precision

Den

sity

00

00

05

01

00

15

Deuterium excesses0 2 4 6 8 10

Paper ID ART20202623 1021275ART20202623 1202

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 5 K-epsilon distribution fit of Monthly Tax Revenue

6 Conclusion

A new distribution called the Kumaraswamy epsilon (K-

epsilon) distribution has been derived in this study It is

constructed from the Kumaraswamy generator

(Kumaraswamy-G) of probability density functions Here

the parent distribution is the epsilon distribution The new

distribution unlike the beta distribution has explicit

expressions for the cumulative distribution and quantile

functions This makes parameter estimates much easier

particularly in R Although the moments of the K-epsilon

distribution could not be expressed in explicit form other

properties of the K-epsilon distribution such as the

distribution of order statistics survival and hazard rate

functions have been derived The shapes of the plots of the

probability density and hazard rate functions at various

parameter values suggest that the K-epsilon distribution

holds the potential for wide applications within its parameter

space

The K-epsilon distribution was applied to two different

lifetime datasets namely deuterium (hydrogen isotope)

excesses and monthly tax revenue The results of the fit of

this distribution to these datasets are good confirming its

wide usefulness in practical applications

References

[1] Aryal G (2015) Kumaraswamy Laplace Distribution

and its Applications Paper presented at the Seventh

International Conference on Dynamic Systems and

Applications and Fifth International Conference on

Neural Parallel and Scientific Computations

[2] Bonne J-L Werner M Meyer H Kipfstuhl S

Rabe B Behrens Melanie K SchAtildeparanicke L amp

Steen-Larsen H-C (2019) Water vapour isotopes

analyser calibrated data from POLARSTERN during

dockyard 2015-08 at Port of TromsAtildecedil Norway

PANGAEA httpsdoiorg101594PANGAEA897410

[3] Bourguignon M Silva R B amp Cordeiro G M (2014)

The Weibull-G family of probability distributions

Journal of Data Science 12 53 ndash 68

[4] Bourguignon M Silva R B Zea L M amp Cordeiro

G M (2013) The Kumaraswamy Pareto distribution

Journal of Statistical Theory and Applications 12 129-

144

[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new

customer lifetime duration distribution the

Kumaraswamy Lindley distribution International

Journal of Trade Economics and Finance Vol 5 No

5 441 ndash 444

[6] Cordeiro G M amp de Castro M (2011) A new family

of generalized distributions Journal of statistical

computation and simulation 81 883-898

httpdxdoiorg10108000949650903530745

[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)

The Kumaraswamy Gumbel distribution Statistical

Methods and Applications 21 139-168

[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27

[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)

The Kumaraswamy Weibull distribution with

application to failure data Journal of the Franklin

Institute Vol 347 No 8 1399-1429

httpdxdoiorg101016jjfranklin201006010

[10] de Pascoa M A R Ortega E M M amp Cordeiro G

M (2011) The Kumaraswamy generalized gamma

distribution with application in survival analysis

Statistical Methodology 8 411ndash433

wwwdoi101016jstamet201104001

[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon

Probability Distribution and its Application in

Reliability Theory Acta PolytechnicaHungarica Vol

15 No 1 pp 197 ndash 216

[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal

distribution and its applications Communication in

Statistics Theory and Methods Vol 31 497-512

[13] Gomes A E da-Silva C Q Cordeiro G M amp

Ortega E M (2014) A new lifetime model the

Kumaraswamy generalized Rayleigh distribution

Journal of statistical computation and simulation 84

290-309

[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001

[15] Klakattawi H S (2019) The Weibull-Gamma

Distribution Properties and Applications Entropy 21

438 pp 1 ndash 15 doi103390e21050438

[16] Kumaraswamy P (1980) A generalized probability

density function for double-bounded random processes

Journal of Hydrology Vol 46 79-88

httpdxdoiorg1010160022-1694(80)90036-0

[17] Nadarajah S (2008) On the distribution of

Kumaraswamy Journal of Hydrology 348 568-569

`httpdxdoiorg101016jjhydrol200709008

[18] Oguntunde P E Babatunde O S ampOgunmola A O

(2014) Theoretical Analysis of the Kumaraswamy-

Inverse Exponential Distribution International Journal

of Statistics and Applications Vol 4 No 2 113-116

doi105923jstatistics2014040205

[19] Zografos K amp Balakrishnan N (2009) On families of

beta and generalized gamma generated distributions and

associated inference Statistical Methodology 6 344 ndash

362

Monthly tax revenue0 10 20 30 40

Densi

ty0

00

00

40

08

00

60

02

Paper ID ART20202623 1021275ART20202623 1203

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

whereΥ = 1205752

1205752minus1199092 Ζ = 1 minus 119909+120575

120575minus119909 minus120582

2120575

0 lt 119909 lt 120575119886 119887 120575 120582 gt

0

The skewness and tail weight of the distribution are

controlled by the parameters 119886 and 119887 The plots of the K-

epsilon density function (8) at various parameter values are

given in Fig 1 below

Figure 1 K-Epsilon probability density plots at various parameter values

From (7) the quantile function of the K-epsilon distribution

is given by

where 0 lt 119901 lt 1 119886 119887 120575 120582 gt 0

Proposition 1 The K-epsilon distribution is a true probability density

function with unimodal shape property That is we want to

show that

(i) 119891119883 119909 is unimodal and

(ii) 119891119883 119909 120575

0119889119909 = 1

The proof of this proposition is given in the appendix

31 Moments of the K-epsilon distribution

The 119903119905119893 119903 = 1 2hellip moment of the K-epsilon can be

derived from the expression

From (10) it is obvious that the moments of the K-epsilon

distribution cannot be obtained in explicit form but can be

obtained by numerical integration for the desired moment(s)

when the parameter values are specified

32 Survival and hazard rate functions

Survival and hazard rate functions form the focal points in

survival analysis and reliability theory The survival

function which is the complementary cumulative

distribution or reliability function is defined as the

probability of surviving to the age of 119909 or exceeding age 119909

On the other hand the hazard rate function also known as

instantaneous failure rate or force of mortality is interpreted

as the amount of risk of failure associated with a unit of age

119909 The two functions are related however the hazard rate

a = 2 b = 04 120575 120582 = 4 (black) 25 (red)

15 (blue) 05 (green)

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

a = 2 120582 = 3 120575 = 6

b = 015 (black) 025 (red)

035 (blue) 05 (green)

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

a = 2 b = 04 120582 120575 = 6 (black) 10 (red)

15 (blue) 25 (green)

0 1 2 3 4 5 6X

values

00

02

04

06

08

Den

sity

0 1 2 3 4 5 6X values

00

02

04

06

08

Den

sity

b = 04 120582 = 3 120575 = 6

a = 2 (black) 3 (red)

5 (blue) 10 (green)

Paper ID ART20202623 1021275ART20202623 1200

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

function is more informative about the underlying

mechanism of failure

The survival function of the K-epsilon distribution is given

by

and its hazard rate function is given by

The plots of the hazard rate function of the K-epsilon

distribution at various parameter values are presented in Fig

2 below The shapes suggest that the K-epsilon hazard rate

function could be a good choice model for somefailure rate

datasets

Figure 2 K-Epsilon distribution hazard rate function at various parameter values

33 Order statistics distribution

Order statistics play important role in life testing and

reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of

a random variable 119883 having the K-epsilon distribution and

density functions 119865119883 and 119891119883 respectively The distribution of

the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip

119909(119903) hellip 119909(119899) is given by

119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)

Where Λ =119899

119903minus1 119899minus119903 Υ =

1205752

1205752minus1199092

Ζ = 1 minus 119909+120575

120575minus119909 minus120582

2120575

andΨ = 1minus Ζ119886 From (13) the K-epsilon

distribution of the first order statistic 119883(1) and the 119899119905119893 order

statistic 119883(119899) are given respectively by

4 Parameter Estimation

As noted earlier the explicit expressions for the K-epsilon

probability density cumulative distribution and quantile

functions has made the distribution a good candidate for the

deployment of the fitdistrplus package in R This has the

advantage of producing parameter estimates their standard

errors and many other important statistical properties

5 Application

The K-epsilon distribution is applied to 2 datasets

namelydeuterium excesses and tax revenue datasets These

are represented in boxplots in Fig 3 below

Paper ID ART20202623 1021275ART20202623 1201

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 3 Boxplots of Deuterium excesses and Monthly Tax

Revenue

The first dataset is hourly record of deuterium (heavy

hydrogen ndash one of the two stable isotopes of hydrogen)

excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]

given in the appendix

Parameter estimates and Kolmogorov-Smirnov goodness-of-

fit test results are given in Table 1 while the fitted K-epsilon

densityplot is presented in Fig 3

Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

From Table 1 it could be observed that the K-epsilon

distribution has fitted the deuterium excesses datasetThe

results of the goodness-of-fit test shown in Table 1 above

provide sufficient evidence that the K-epsilon distribution

fits the dataset Also the standard errors of the parameter

estimates obtained indicate they are of good precision

Figure 3 K-Epsilon distribution fit of hydrogen

Isotope (Deuterium) excesses

The second dataset for application is the monthly records of

tax revenue of Egypt between January 2006 and November

2010 in 1000 million Egyptian pounds reported in [15] It is

presented in the appendix

The results for parameter estimate and goodness-of-fit test

are presented in Table 2 while the fitted K-epsilon

probability density plot is presented in Fig 4

Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

Table 2 also suggest that the K-epsilon distribution has

successfully fitted the tax revenue dataset The parameter

estimates are of good precision

Den

sity

00

00

05

01

00

15

Deuterium excesses0 2 4 6 8 10

Paper ID ART20202623 1021275ART20202623 1202

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 5 K-epsilon distribution fit of Monthly Tax Revenue

6 Conclusion

A new distribution called the Kumaraswamy epsilon (K-

epsilon) distribution has been derived in this study It is

constructed from the Kumaraswamy generator

(Kumaraswamy-G) of probability density functions Here

the parent distribution is the epsilon distribution The new

distribution unlike the beta distribution has explicit

expressions for the cumulative distribution and quantile

functions This makes parameter estimates much easier

particularly in R Although the moments of the K-epsilon

distribution could not be expressed in explicit form other

properties of the K-epsilon distribution such as the

distribution of order statistics survival and hazard rate

functions have been derived The shapes of the plots of the

probability density and hazard rate functions at various

parameter values suggest that the K-epsilon distribution

holds the potential for wide applications within its parameter

space

The K-epsilon distribution was applied to two different

lifetime datasets namely deuterium (hydrogen isotope)

excesses and monthly tax revenue The results of the fit of

this distribution to these datasets are good confirming its

wide usefulness in practical applications

References

[1] Aryal G (2015) Kumaraswamy Laplace Distribution

and its Applications Paper presented at the Seventh

International Conference on Dynamic Systems and

Applications and Fifth International Conference on

Neural Parallel and Scientific Computations

[2] Bonne J-L Werner M Meyer H Kipfstuhl S

Rabe B Behrens Melanie K SchAtildeparanicke L amp

Steen-Larsen H-C (2019) Water vapour isotopes

analyser calibrated data from POLARSTERN during

dockyard 2015-08 at Port of TromsAtildecedil Norway

PANGAEA httpsdoiorg101594PANGAEA897410

[3] Bourguignon M Silva R B amp Cordeiro G M (2014)

The Weibull-G family of probability distributions

Journal of Data Science 12 53 ndash 68

[4] Bourguignon M Silva R B Zea L M amp Cordeiro

G M (2013) The Kumaraswamy Pareto distribution

Journal of Statistical Theory and Applications 12 129-

144

[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new

customer lifetime duration distribution the

Kumaraswamy Lindley distribution International

Journal of Trade Economics and Finance Vol 5 No

5 441 ndash 444

[6] Cordeiro G M amp de Castro M (2011) A new family

of generalized distributions Journal of statistical

computation and simulation 81 883-898

httpdxdoiorg10108000949650903530745

[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)

The Kumaraswamy Gumbel distribution Statistical

Methods and Applications 21 139-168

[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27

[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)

The Kumaraswamy Weibull distribution with

application to failure data Journal of the Franklin

Institute Vol 347 No 8 1399-1429

httpdxdoiorg101016jjfranklin201006010

[10] de Pascoa M A R Ortega E M M amp Cordeiro G

M (2011) The Kumaraswamy generalized gamma

distribution with application in survival analysis

Statistical Methodology 8 411ndash433

wwwdoi101016jstamet201104001

[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon

Probability Distribution and its Application in

Reliability Theory Acta PolytechnicaHungarica Vol

15 No 1 pp 197 ndash 216

[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal

distribution and its applications Communication in

Statistics Theory and Methods Vol 31 497-512

[13] Gomes A E da-Silva C Q Cordeiro G M amp

Ortega E M (2014) A new lifetime model the

Kumaraswamy generalized Rayleigh distribution

Journal of statistical computation and simulation 84

290-309

[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001

[15] Klakattawi H S (2019) The Weibull-Gamma

Distribution Properties and Applications Entropy 21

438 pp 1 ndash 15 doi103390e21050438

[16] Kumaraswamy P (1980) A generalized probability

density function for double-bounded random processes

Journal of Hydrology Vol 46 79-88

httpdxdoiorg1010160022-1694(80)90036-0

[17] Nadarajah S (2008) On the distribution of

Kumaraswamy Journal of Hydrology 348 568-569

`httpdxdoiorg101016jjhydrol200709008

[18] Oguntunde P E Babatunde O S ampOgunmola A O

(2014) Theoretical Analysis of the Kumaraswamy-

Inverse Exponential Distribution International Journal

of Statistics and Applications Vol 4 No 2 113-116

doi105923jstatistics2014040205

[19] Zografos K amp Balakrishnan N (2009) On families of

beta and generalized gamma generated distributions and

associated inference Statistical Methodology 6 344 ndash

362

Monthly tax revenue0 10 20 30 40

Densi

ty0

00

00

40

08

00

60

02

Paper ID ART20202623 1021275ART20202623 1203

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

function is more informative about the underlying

mechanism of failure

The survival function of the K-epsilon distribution is given

by

and its hazard rate function is given by

The plots of the hazard rate function of the K-epsilon

distribution at various parameter values are presented in Fig

2 below The shapes suggest that the K-epsilon hazard rate

function could be a good choice model for somefailure rate

datasets

Figure 2 K-Epsilon distribution hazard rate function at various parameter values

33 Order statistics distribution

Order statistics play important role in life testing and

reliability analysis Given a random sample 1199091 1199092 hellip 119909119899 of

a random variable 119883 having the K-epsilon distribution and

density functions 119865119883 and 119891119883 respectively The distribution of

the 119903119905119893order statistic 119883(119903) of the ordered sample 119909(1) hellip

119909(119903) hellip 119909(119899) is given by

119891119883 119903 119909 = 119886119887120582ΛΥ 1 minus Ζ Ζ119886minus1Ψ119887 119899minus119903+1 minus1 1 minusΨ119887 119903minus1(13)

Where Λ =119899

119903minus1 119899minus119903 Υ =

1205752

1205752minus1199092

Ζ = 1 minus 119909+120575

120575minus119909 minus120582

2120575

andΨ = 1minus Ζ119886 From (13) the K-epsilon

distribution of the first order statistic 119883(1) and the 119899119905119893 order

statistic 119883(119899) are given respectively by

4 Parameter Estimation

As noted earlier the explicit expressions for the K-epsilon

probability density cumulative distribution and quantile

functions has made the distribution a good candidate for the

deployment of the fitdistrplus package in R This has the

advantage of producing parameter estimates their standard

errors and many other important statistical properties

5 Application

The K-epsilon distribution is applied to 2 datasets

namelydeuterium excesses and tax revenue datasets These

are represented in boxplots in Fig 3 below

Paper ID ART20202623 1021275ART20202623 1201

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 3 Boxplots of Deuterium excesses and Monthly Tax

Revenue

The first dataset is hourly record of deuterium (heavy

hydrogen ndash one of the two stable isotopes of hydrogen)

excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]

given in the appendix

Parameter estimates and Kolmogorov-Smirnov goodness-of-

fit test results are given in Table 1 while the fitted K-epsilon

densityplot is presented in Fig 3

Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

From Table 1 it could be observed that the K-epsilon

distribution has fitted the deuterium excesses datasetThe

results of the goodness-of-fit test shown in Table 1 above

provide sufficient evidence that the K-epsilon distribution

fits the dataset Also the standard errors of the parameter

estimates obtained indicate they are of good precision

Figure 3 K-Epsilon distribution fit of hydrogen

Isotope (Deuterium) excesses

The second dataset for application is the monthly records of

tax revenue of Egypt between January 2006 and November

2010 in 1000 million Egyptian pounds reported in [15] It is

presented in the appendix

The results for parameter estimate and goodness-of-fit test

are presented in Table 2 while the fitted K-epsilon

probability density plot is presented in Fig 4

Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

Table 2 also suggest that the K-epsilon distribution has

successfully fitted the tax revenue dataset The parameter

estimates are of good precision

Den

sity

00

00

05

01

00

15

Deuterium excesses0 2 4 6 8 10

Paper ID ART20202623 1021275ART20202623 1202

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 5 K-epsilon distribution fit of Monthly Tax Revenue

6 Conclusion

A new distribution called the Kumaraswamy epsilon (K-

epsilon) distribution has been derived in this study It is

constructed from the Kumaraswamy generator

(Kumaraswamy-G) of probability density functions Here

the parent distribution is the epsilon distribution The new

distribution unlike the beta distribution has explicit

expressions for the cumulative distribution and quantile

functions This makes parameter estimates much easier

particularly in R Although the moments of the K-epsilon

distribution could not be expressed in explicit form other

properties of the K-epsilon distribution such as the

distribution of order statistics survival and hazard rate

functions have been derived The shapes of the plots of the

probability density and hazard rate functions at various

parameter values suggest that the K-epsilon distribution

holds the potential for wide applications within its parameter

space

The K-epsilon distribution was applied to two different

lifetime datasets namely deuterium (hydrogen isotope)

excesses and monthly tax revenue The results of the fit of

this distribution to these datasets are good confirming its

wide usefulness in practical applications

References

[1] Aryal G (2015) Kumaraswamy Laplace Distribution

and its Applications Paper presented at the Seventh

International Conference on Dynamic Systems and

Applications and Fifth International Conference on

Neural Parallel and Scientific Computations

[2] Bonne J-L Werner M Meyer H Kipfstuhl S

Rabe B Behrens Melanie K SchAtildeparanicke L amp

Steen-Larsen H-C (2019) Water vapour isotopes

analyser calibrated data from POLARSTERN during

dockyard 2015-08 at Port of TromsAtildecedil Norway

PANGAEA httpsdoiorg101594PANGAEA897410

[3] Bourguignon M Silva R B amp Cordeiro G M (2014)

The Weibull-G family of probability distributions

Journal of Data Science 12 53 ndash 68

[4] Bourguignon M Silva R B Zea L M amp Cordeiro

G M (2013) The Kumaraswamy Pareto distribution

Journal of Statistical Theory and Applications 12 129-

144

[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new

customer lifetime duration distribution the

Kumaraswamy Lindley distribution International

Journal of Trade Economics and Finance Vol 5 No

5 441 ndash 444

[6] Cordeiro G M amp de Castro M (2011) A new family

of generalized distributions Journal of statistical

computation and simulation 81 883-898

httpdxdoiorg10108000949650903530745

[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)

The Kumaraswamy Gumbel distribution Statistical

Methods and Applications 21 139-168

[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27

[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)

The Kumaraswamy Weibull distribution with

application to failure data Journal of the Franklin

Institute Vol 347 No 8 1399-1429

httpdxdoiorg101016jjfranklin201006010

[10] de Pascoa M A R Ortega E M M amp Cordeiro G

M (2011) The Kumaraswamy generalized gamma

distribution with application in survival analysis

Statistical Methodology 8 411ndash433

wwwdoi101016jstamet201104001

[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon

Probability Distribution and its Application in

Reliability Theory Acta PolytechnicaHungarica Vol

15 No 1 pp 197 ndash 216

[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal

distribution and its applications Communication in

Statistics Theory and Methods Vol 31 497-512

[13] Gomes A E da-Silva C Q Cordeiro G M amp

Ortega E M (2014) A new lifetime model the

Kumaraswamy generalized Rayleigh distribution

Journal of statistical computation and simulation 84

290-309

[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001

[15] Klakattawi H S (2019) The Weibull-Gamma

Distribution Properties and Applications Entropy 21

438 pp 1 ndash 15 doi103390e21050438

[16] Kumaraswamy P (1980) A generalized probability

density function for double-bounded random processes

Journal of Hydrology Vol 46 79-88

httpdxdoiorg1010160022-1694(80)90036-0

[17] Nadarajah S (2008) On the distribution of

Kumaraswamy Journal of Hydrology 348 568-569

`httpdxdoiorg101016jjhydrol200709008

[18] Oguntunde P E Babatunde O S ampOgunmola A O

(2014) Theoretical Analysis of the Kumaraswamy-

Inverse Exponential Distribution International Journal

of Statistics and Applications Vol 4 No 2 113-116

doi105923jstatistics2014040205

[19] Zografos K amp Balakrishnan N (2009) On families of

beta and generalized gamma generated distributions and

associated inference Statistical Methodology 6 344 ndash

362

Monthly tax revenue0 10 20 30 40

Densi

ty0

00

00

40

08

00

60

02

Paper ID ART20202623 1021275ART20202623 1203

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 3 Boxplots of Deuterium excesses and Monthly Tax

Revenue

The first dataset is hourly record of deuterium (heavy

hydrogen ndash one of the two stable isotopes of hydrogen)

excesses from 2015-11-11T0000 to 2015-11-24T2300 [2]

given in the appendix

Parameter estimates and Kolmogorov-Smirnov goodness-of-

fit test results are given in Table 1 while the fitted K-epsilon

densityplot is presented in Fig 3

Table 1 K-epsilon distribution fit results for deuterium excesses dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 15866 (01061) 22124 (959727) 00050 (00007) 97456 (02744) 67668 00518 (00781) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

From Table 1 it could be observed that the K-epsilon

distribution has fitted the deuterium excesses datasetThe

results of the goodness-of-fit test shown in Table 1 above

provide sufficient evidence that the K-epsilon distribution

fits the dataset Also the standard errors of the parameter

estimates obtained indicate they are of good precision

Figure 3 K-Epsilon distribution fit of hydrogen

Isotope (Deuterium) excesses

The second dataset for application is the monthly records of

tax revenue of Egypt between January 2006 and November

2010 in 1000 million Egyptian pounds reported in [15] It is

presented in the appendix

The results for parameter estimate and goodness-of-fit test

are presented in Table 2 while the fitted K-epsilon

probability density plot is presented in Fig 4

Table 2 K-epsilon distribution fit results for Monthly Tax Revenue dataset Distribution 119886 (119904119890) 119887 119904119890 120582 119904119890 120575 (119904119890) minus119871119871 119870119878(119862119881) Remark

K-epsilon 638712 (18779) 01877 (00378) 06924 (00637) 20488 (02086) 187973 00735 (01771) Good fit

119904119890 = standard error LL = log-likelihood KS = Kolmogorov-Smirnov 119862119881 = critical value

Table 2 also suggest that the K-epsilon distribution has

successfully fitted the tax revenue dataset The parameter

estimates are of good precision

Den

sity

00

00

05

01

00

15

Deuterium excesses0 2 4 6 8 10

Paper ID ART20202623 1021275ART20202623 1202

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 5 K-epsilon distribution fit of Monthly Tax Revenue

6 Conclusion

A new distribution called the Kumaraswamy epsilon (K-

epsilon) distribution has been derived in this study It is

constructed from the Kumaraswamy generator

(Kumaraswamy-G) of probability density functions Here

the parent distribution is the epsilon distribution The new

distribution unlike the beta distribution has explicit

expressions for the cumulative distribution and quantile

functions This makes parameter estimates much easier

particularly in R Although the moments of the K-epsilon

distribution could not be expressed in explicit form other

properties of the K-epsilon distribution such as the

distribution of order statistics survival and hazard rate

functions have been derived The shapes of the plots of the

probability density and hazard rate functions at various

parameter values suggest that the K-epsilon distribution

holds the potential for wide applications within its parameter

space

The K-epsilon distribution was applied to two different

lifetime datasets namely deuterium (hydrogen isotope)

excesses and monthly tax revenue The results of the fit of

this distribution to these datasets are good confirming its

wide usefulness in practical applications

References

[1] Aryal G (2015) Kumaraswamy Laplace Distribution

and its Applications Paper presented at the Seventh

International Conference on Dynamic Systems and

Applications and Fifth International Conference on

Neural Parallel and Scientific Computations

[2] Bonne J-L Werner M Meyer H Kipfstuhl S

Rabe B Behrens Melanie K SchAtildeparanicke L amp

Steen-Larsen H-C (2019) Water vapour isotopes

analyser calibrated data from POLARSTERN during

dockyard 2015-08 at Port of TromsAtildecedil Norway

PANGAEA httpsdoiorg101594PANGAEA897410

[3] Bourguignon M Silva R B amp Cordeiro G M (2014)

The Weibull-G family of probability distributions

Journal of Data Science 12 53 ndash 68

[4] Bourguignon M Silva R B Zea L M amp Cordeiro

G M (2013) The Kumaraswamy Pareto distribution

Journal of Statistical Theory and Applications 12 129-

144

[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new

customer lifetime duration distribution the

Kumaraswamy Lindley distribution International

Journal of Trade Economics and Finance Vol 5 No

5 441 ndash 444

[6] Cordeiro G M amp de Castro M (2011) A new family

of generalized distributions Journal of statistical

computation and simulation 81 883-898

httpdxdoiorg10108000949650903530745

[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)

The Kumaraswamy Gumbel distribution Statistical

Methods and Applications 21 139-168

[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27

[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)

The Kumaraswamy Weibull distribution with

application to failure data Journal of the Franklin

Institute Vol 347 No 8 1399-1429

httpdxdoiorg101016jjfranklin201006010

[10] de Pascoa M A R Ortega E M M amp Cordeiro G

M (2011) The Kumaraswamy generalized gamma

distribution with application in survival analysis

Statistical Methodology 8 411ndash433

wwwdoi101016jstamet201104001

[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon

Probability Distribution and its Application in

Reliability Theory Acta PolytechnicaHungarica Vol

15 No 1 pp 197 ndash 216

[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal

distribution and its applications Communication in

Statistics Theory and Methods Vol 31 497-512

[13] Gomes A E da-Silva C Q Cordeiro G M amp

Ortega E M (2014) A new lifetime model the

Kumaraswamy generalized Rayleigh distribution

Journal of statistical computation and simulation 84

290-309

[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001

[15] Klakattawi H S (2019) The Weibull-Gamma

Distribution Properties and Applications Entropy 21

438 pp 1 ndash 15 doi103390e21050438

[16] Kumaraswamy P (1980) A generalized probability

density function for double-bounded random processes

Journal of Hydrology Vol 46 79-88

httpdxdoiorg1010160022-1694(80)90036-0

[17] Nadarajah S (2008) On the distribution of

Kumaraswamy Journal of Hydrology 348 568-569

`httpdxdoiorg101016jjhydrol200709008

[18] Oguntunde P E Babatunde O S ampOgunmola A O

(2014) Theoretical Analysis of the Kumaraswamy-

Inverse Exponential Distribution International Journal

of Statistics and Applications Vol 4 No 2 113-116

doi105923jstatistics2014040205

[19] Zografos K amp Balakrishnan N (2009) On families of

beta and generalized gamma generated distributions and

associated inference Statistical Methodology 6 344 ndash

362

Monthly tax revenue0 10 20 30 40

Densi

ty0

00

00

40

08

00

60

02

Paper ID ART20202623 1021275ART20202623 1203

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Figure 5 K-epsilon distribution fit of Monthly Tax Revenue

6 Conclusion

A new distribution called the Kumaraswamy epsilon (K-

epsilon) distribution has been derived in this study It is

constructed from the Kumaraswamy generator

(Kumaraswamy-G) of probability density functions Here

the parent distribution is the epsilon distribution The new

distribution unlike the beta distribution has explicit

expressions for the cumulative distribution and quantile

functions This makes parameter estimates much easier

particularly in R Although the moments of the K-epsilon

distribution could not be expressed in explicit form other

properties of the K-epsilon distribution such as the

distribution of order statistics survival and hazard rate

functions have been derived The shapes of the plots of the

probability density and hazard rate functions at various

parameter values suggest that the K-epsilon distribution

holds the potential for wide applications within its parameter

space

The K-epsilon distribution was applied to two different

lifetime datasets namely deuterium (hydrogen isotope)

excesses and monthly tax revenue The results of the fit of

this distribution to these datasets are good confirming its

wide usefulness in practical applications

References

[1] Aryal G (2015) Kumaraswamy Laplace Distribution

and its Applications Paper presented at the Seventh

International Conference on Dynamic Systems and

Applications and Fifth International Conference on

Neural Parallel and Scientific Computations

[2] Bonne J-L Werner M Meyer H Kipfstuhl S

Rabe B Behrens Melanie K SchAtildeparanicke L amp

Steen-Larsen H-C (2019) Water vapour isotopes

analyser calibrated data from POLARSTERN during

dockyard 2015-08 at Port of TromsAtildecedil Norway

PANGAEA httpsdoiorg101594PANGAEA897410

[3] Bourguignon M Silva R B amp Cordeiro G M (2014)

The Weibull-G family of probability distributions

Journal of Data Science 12 53 ndash 68

[4] Bourguignon M Silva R B Zea L M amp Cordeiro

G M (2013) The Kumaraswamy Pareto distribution

Journal of Statistical Theory and Applications 12 129-

144

[5] Ccedilakmakyapan S ampKadilar G Ouml (2014) A new

customer lifetime duration distribution the

Kumaraswamy Lindley distribution International

Journal of Trade Economics and Finance Vol 5 No

5 441 ndash 444

[6] Cordeiro G M amp de Castro M (2011) A new family

of generalized distributions Journal of statistical

computation and simulation 81 883-898

httpdxdoiorg10108000949650903530745

[7] Cordeiro G M Nadarajah S amp Ortega E M (2012)

The Kumaraswamy Gumbel distribution Statistical

Methods and Applications 21 139-168

[8] Cordeiro G M Ortega E Mamp da Cunha D C C (2013) The exponentiated generalized class of distributions Journal of Data Science 11 1 ndash 27

[9] Cordeiro G M Ortega E M amp Nadarajah S (2010)

The Kumaraswamy Weibull distribution with

application to failure data Journal of the Franklin

Institute Vol 347 No 8 1399-1429

httpdxdoiorg101016jjfranklin201006010

[10] de Pascoa M A R Ortega E M M amp Cordeiro G

M (2011) The Kumaraswamy generalized gamma

distribution with application in survival analysis

Statistical Methodology 8 411ndash433

wwwdoi101016jstamet201104001

[11] Dombi J Jacuteonacuteas T ampTacuteoth Z E (2018) The Epsilon

Probability Distribution and its Application in

Reliability Theory Acta PolytechnicaHungarica Vol

15 No 1 pp 197 ndash 216

[12] Eugene N Lee C ampFamoye F (2002) Beta-Normal

distribution and its applications Communication in

Statistics Theory and Methods Vol 31 497-512

[13] Gomes A E da-Silva C Q Cordeiro G M amp

Ortega E M (2014) A new lifetime model the

Kumaraswamy generalized Rayleigh distribution

Journal of statistical computation and simulation 84

290-309

[14] Jones M C (2009) Kumaraswamyrsquos distribution ldquoA beta-type distribution with some tractability advantagesrdquo Statistical Methodology 6 70-81 httpdxdoiorg101016jstamet200804001

[15] Klakattawi H S (2019) The Weibull-Gamma

Distribution Properties and Applications Entropy 21

438 pp 1 ndash 15 doi103390e21050438

[16] Kumaraswamy P (1980) A generalized probability

density function for double-bounded random processes

Journal of Hydrology Vol 46 79-88

httpdxdoiorg1010160022-1694(80)90036-0

[17] Nadarajah S (2008) On the distribution of

Kumaraswamy Journal of Hydrology 348 568-569

`httpdxdoiorg101016jjhydrol200709008

[18] Oguntunde P E Babatunde O S ampOgunmola A O

(2014) Theoretical Analysis of the Kumaraswamy-

Inverse Exponential Distribution International Journal

of Statistics and Applications Vol 4 No 2 113-116

doi105923jstatistics2014040205

[19] Zografos K amp Balakrishnan N (2009) On families of

beta and generalized gamma generated distributions and

associated inference Statistical Methodology 6 344 ndash

362

Monthly tax revenue0 10 20 30 40

Densi

ty0

00

00

40

08

00

60

02

Paper ID ART20202623 1021275ART20202623 1203

International Journal of Science and Research (IJSR) ISSN 2319-7064

ResearchGate Impact Factor (2018) 028 | SJIF (2018) 7426

Volume 8 Issue 11 November 2019

wwwijsrnet Licensed Under Creative Commons Attribution CC BY

Appendix

Proof of Proposition 1

Proof

(i) We need to find the limiting values of 119891119883 119909 That is

lim119909rarr0

119891119883 119909 = lim119909rarr0

119886119887120582 1205752

1205752 minus 1199092 119909 + 120575

120575 minus 119909 minus120582

2120575

1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886minus1

1 minus 1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887minus1

= 119886119887120582 lim119909rarr0

1205752

1205752 minus 1199092 lim119909rarr0

119909 + 120575

120575 minus 119909 minus120582

2120575

lim119909rarr0

1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886minus1

lim119909rarr0

1minus 1 minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887minus1

= 119886119887120582 ∙ 1 ∙ 1 ∙ 1 minus 1 119886minus1 ∙ 1 minus 1 minus 1 119886 119887minus1

= 0

Also

lim119909rarr120575

119891119883 119909 = lim119909rarr120575

119886119887120582 1205752

1205752 minus 1199092 119909 + 120575

120575 minus 119909 minus120582

2120575

1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886minus1

1 minus 1 minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887minus1

= 119886119887120582 lim119909rarr120575

1205752

1205752 minus 1199092 lim119909rarr120575

119909 + 120575

120575 minus 119909 minus120582

2120575

lim119909rarr120575

1 minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886minus1

lim119909rarr120575

1minus 1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887minus1

= 119886119887120582 ∙ infin ∙ 0 ∙ 1 minus 0 119886minus1 ∙ 1 minus 1 minus 0 119886 119887minus1

= 0

Since 119891119883 119909 at both limits is zero (0) it is unimodal

(ii) 119891119883 119909 120575

0119889119909 = lim119909rarr120575 119891119883 119905

119909

0119889119905

= lim119909rarr120575

119865119883 119909

= lim119909rarr120575

1minus 1 minus 1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887

= 1 minus lim119909rarr120575

1 minus 1minus 119909 + 120575

120575 minus 119909 minus120582

2120575

119886

119887

= 1 minus 1minus 1minus 120575 + 120575

120575 minus 120575 minus120582

2120575

119886

119887

= 1minus 1 minus 1 minus 0 119886 119887

= 1

Hourly record of deuterium excesses from 2015-11-11T0000 to 2015-11-24T2300

302 257 174 225 155 066 056 092 124 120 086 118 134 194 197 179 182 194 206 184 224 201

232 315 345 338 354 389 498 343 398 419 436 497 465 415 349 319 337 358 382 400 421 469

486 506 493 477 494 495 534 526 476 535 559 554 578 592 580 569 585 574 564 582 521 548

619 792 893 845 874 835 840 800 769 622 699 685 745 788 780 805 822 840 790 665 629 674

572 475 412 420 386 352 353 327 323 313 262 394 452 512 577 594 592 586 586 579 535 557

571 581 596 604 658 669 670 697 699 723 732 661 720 799 817 850 858 851 897 888 873 862

893 945 912 909 873 898 876 891 918 930 923 907 859 908 947 912 873 847 889 815 677 672

641 617 479 470 435 432 432 411 398 425 421 397 351 270 256 267 212 258 237 199 157 120

119 070 094 107 109 131 118 086 081 047 070 083 104 169 065 133 155 213 238 267 249 219

220 169 166 187 267 243 193 263 307 361 340 314 354 293 339 245 250 341 329 357 341 367

303 337 371 374 384 352 347 375 365 348 349 440 367 441 365 349 285 379 386 400 388 469

427 380 372 389 371 293 210 179 298 391 528 551 583 554 542 493 450 312 433 460 400 430

485 501 490 550 607 636 650 683 739 756 722 728 787 763 768 700 728 719 686 691 664672

687 714 686 680 630 623 590 602 616 585 598 555 553 545 561 531 524 492 527 224 250 257

Monthly records of tax revenue of Egypt between January 2006 and November 2010 (1000 million Egyptian pounds)

59 204 149 162 172 78 61 92 102 96 133 85 216 185 51 67 17 86 97 392 357 157 97 1041 36

85 8 92 262 219 167 213 354 143 85 106 191 205 71 77 181 165 119 7 86 125103 112 61 84

11 116 119 52 68 89 7110

Paper ID ART20202623 1021275ART20202623 1204