Open access peer-reviewed chapter

# Gamma-Kumaraswamy Distribution in Reliability Analysis: Properties and Applications

Written By

Indranil Ghosh and Gholamhossein G. Hamedani

Submitted: April 14th, 2016 Reviewed: November 9th, 2016 Published: April 26th, 2017

DOI: 10.5772/66821

From the Edited Volume

## Advances in Statistical Methodologies and Their Application to Real Problems

Edited by Tsukasa Hokimoto

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## Abstract

In this chapter, a new generalization of the Kumaraswamy distribution, namely the gamma-Kumaraswamy distribution is defined and studied. Several distributional properties of the distribution are discussed in this chapter, which includes limiting behavior, mode, quantiles, moments, skewness, kurtosis, Shannon’s entropy, and order statistics. Under the classical method of estimation, the method of maximum likelihood estimation is proposed for the inference of this distribution. We provide the results of an analysis based on two real data sets when applied to the gamma-Kumaraswamy distribution to exhibit the utility of this model.

### Keywords

• gamma-Kumaraswamy distribution
• Renyi’s entropy
• reliability parameter
• stochastic ordering
• characterizations

## 1. Introduction

The generalization of a distribution by mixing it with another distribution over the years has provided a mathematical based way to model a wide variety of random phenomena statistically. These generalized distributions are effective and flexible models to analyze and interpret random durations in a possibly heterogeneous population. In many situations, observed data may be assumed to have come from such a mixture population of two or more distributions.

Two parameter gamma and a two parameter Kumaraswamy are most popular distribution for analyzing any lifetime data. Gamma distribution is a well-known distribution, and it has several desirable properties [1].

A serious limitation of the gamma distribution, however, is that the distribution function (or survival function) is not available in a closed form if the shape parameter is not an integer, thereby it requires some numerical methods to evaluate these quantities. As a consequence, this distribution is less attractive as compared to Ref. [2], which has nice tractable distribution function, survival function and hazard function. In this paper, we consider a four parameter gamma-Kumaraswamy distribution. It is observed that it has many properties which are quite similar to those of a gamma distribution, but it has an explicit expression for the distribution function or the survival functions. The major motivation of this chapter is to introduce a new family of distributions, make a comparative study of this family with respect to a Kumaraswamy family and a gamma family and provide the practitioner with an additional option, with a hope that it may have a ‘better fit’ compared to a gamma family or Kumaraswamy family in certain situations. It is noteworthy to note that the gamma-Kumaraswamy distribution is a generalization of Kumaraswamy distribution with the property that it can exhibit various shapes. ( Figure 1 ). This provides more flexibility to the gamma-Kumaraswamy distribution in comparison with Kumaraswamy distribution in modeling different data sets. The property of left-skewness is a rare characteristic as it is not enjoyed by several generalizations of Kumaraswamy distribution. Our proposed model is different from that of Ref. [3], where the authors have proposed a generalized gamma-generated distribution with an extra positive parameter for any continuous baseline Gdistribution.

The rest of the paper is organized as follows. In Section 2, we propose the gamma-Kumaraswamy distribution [GK(α, β, a, b)]. In Section 3, we study various properties of the GK(α, β, a, b) including the limiting behavior, transformation, and the mode. In Section 4, the moment generating function, the moments and the mean deviations from the mean and the median, and Renyi’s entropy are studied. In Section 5, we consider the maximum likelihood estimation of the GK(α, β, a, b). In Section 6, we provide an expression for the reliability parameter for two independent GK(α, β, a, b) with different choices for the parameters αand βbut for a fixed choice of the two shape parameters of Kumaraswamy distribution. In Section 7, discussion is made for the moment generating function of the r-th order statistic and also the limiting distribution of the sample minimum and the sample maximum for a random sample of size ndrawn from GK(α, β, a, b). An application of GK(α, β, a, b) is discussed in Section 8. Certain characterizations of GK(α, β, a, b) are presented in Section 9. In Section 10, some concluding remarks are made.

## 2. The gamma-Kumaraswamy distribution

We consider the following class of gamma-Xclass of distributions, for which, the parent model being

f(x)=1Γ(α)βαg(x)G¯2(x)exp(g(x)βG¯(x))(G(x)G¯(x))α1,x>0,E1

where α, βare positive parameters. Also, g(x)[G(x)]is the density function [cumulative distribution function] of the random variable X. Furthermore, G¯(x)is the survival function of the associated random variable X.

If Xhas density Eq. (1), then the random variable W=G(x)G¯(x)has a gamma distribution with parameters α, β. The reverse happens to be true as well. Here, we consider G(.) to be the cdf of a Kumaraswamy distribution with parameters a, b. Then, the cdf of the gamma-Kumaraswamy (hereafter GK) reduces to

F(x)=01(1xa)b(1xa)bew/βwα1Γ(α)βαdw=γ1(α,1(1xa)bβ(1xa)b),0<x<1.E2

where γ1(α,z)=Γ(α,z)Γ(α)with Γ(α,x)=0xuα1euduis the regularized incomplete gamma function. So the density and hazard functions corresponding to Eq. (2) are given, respectively, by

f(x)=abexp(1(1xa)bβ(1xa)b)Γ(α)βα1(1xa)2(1(1xa)bβ(1xa)b)α1xa1,0<x<1,E3

and

hF(x)=((1xa)b1)α1abxa1(1xa)b1exp(β1((1xa)b1))βα(1xa)b+1(1Γ(α,1(1xa)bβ(1xa)b)).E4

The percentile functions for GK distribution: The pth percentile xpis defined by F(xp) = p. From Eq. (2), we have γ1(α,1(1xa)bβ(1xa)b)=p. Define Zp=1(1xa)bβ(1xa)b, then Zp=γ11(α,p), where γ11is the inverse of regularized incomplete gamma function. Hence, xp=(1(β(1+Z1p))1/b)1/a.

In the density equation (3), a, b, and αare shape parameters and βis the scale parameter. It can be immediately verified that Eq. (3) is a density function. Plots of the GK density and survival rate function for selected parameter values are given in Figures 1 and 2 , respectively.

If X~GK(a, b, α, β), then the survival function of X, S(x) will be

1γ1(α,1(1xa)bβ(1xa)b).E5

We simulate the GK distribution by solving the nonlinear equation

(1u)γ1(α,1(1xa)bβ(1xa)b)=0,E6

where uhas the uniform (0,1) distribution. Some facts regarding the GK distribution are as follows:

• If X~GK(a, b, α, β), then Xm~GK(a, b, α, β), m0.

• Also, we have the following important result: If X~GK(1, b, α, β), then X1/a~GK(a, b, α, β), a0.

• The GK distribution does not possess the reproductive property. In other words, if for any two X1~GK(a1,b1,α1,β1)and X2~GK(a2,b2,α2,β2),then the distribution of the sum S= X1 + X2 will not be a GK.

The first result provides an important property of the GK distribution for information analysis is that this distribution is closed under power transformation. The latter result is equally important because it provides a simple way to generate random variables following the GK distribution.

## 3. Properties of GK distribution

The following lemma establishes the relation between GK(α, β, a, b) distribution and gamma distribution.

Lemma 1. (Transformation): If a random variable Xfollows a gamma distribution with parameters αand β, then Y=1(1Xa)b(1Xa)bfollows GK(α, β, a, b) distribution.

Proof. The proof follows immediately by using the transformation technique. W

The limiting behaviors of the GK pdf and its hazard function are given in the following theorem.

Theorem 1.The limits GK density function, f(x), and the hazard function, hF(x), are given by

limx0+f(x)=limx0+hf(x)={0,a>1,b>1,α>1,min{a,b}<1,α<1,E7
limxf(x)=limxhf(x)={0,b>0,α<1,b<0,α>1.E8

Proof. Straightforward and hence omitted. W

Theorem 2.The mode of the GK distribution is the solution of the equation k(x)=0,where

k(x)=(a1)2xa(1xa)+abxa(1xa)b(β1+(1(1xa)bβ(1xa)b)1).E9

Proof. The derivative of f(x) in Eq. (3) can be written as

xf(x)=1βαΓ(α)abxa2(1xa)2exp(β1((1xa)b1))(1(1xa)bβ(1xa)b)α1k(x).E10

The critical values of Eq. (10) are the solutions of k(x)=0.W

Next, we discuss the IFR and/or DFR property of the hazard function for the GK distribution. For this, we will consider the result of Lemma 1. According to Lemma 1, if X~GK(a, b, α, β), then Y=1(1Xa)b(1Xa)bGamma(α, β). In such a case for the random variable Y, the hazard rate function can be written as

1r(t)=1F(t)f(t)=t1βαΓ(α)wα1exp(w/β)dw1βαΓ(α)tα1exp(t/%beta)=tinfty(wt)α1exp(1/β(wt))dw=0(1+ut)α1exp(1/βu)du.E11

Therefore, r(t)=(0(1+ut)α1exp(1/βu)du)1. If α>1, (1+ut)α1is decreasing in tand hence r(t) is increasing, thereby and has a IFR. If 0<α<1, then

(1+ut)α1is increasing in t, so r(t) decreases and hence has a DFR. Now, since Xis a one-to-one function of Y, the hazard rate function of Xwill also follow the exact pattern.

Let Xand Ybe two random variables. Xis said to be stochastically greater than or equal to Ydenoted by XstYif P(X>x)P(Y>x)for all xin the support set of X.

Theorem 3.Suppose X~GK(a1,b1,α,β1)and Y~GK(a2,b2,α,β2).If β1 > β2, a1 > a2 and b1 < b2. Then XstY, for integer values of a1 and a2.

Proof. At first, we note that the incomplete gamma function Γ(α,x)is an increasing function of xfor fixed α. For any real number x(0,1),β1>β2,a1>a2, and b1<b2, we have

β11((1xa1)b11)β21((1xa2)b21).E12

This implies that Γ(α,β11((1xa1)b11))Γ(α,β21((1xa2)b21)). Equivalently, it implies that P(X>x)P(Y>x), and this completes the proof. W

Note:For fractional choices of a1 and a2, the reverse of the above inequality will hold.

## 4. Moments and mean deviations

For any r1,

Upper bounds for the r-th order moment: Since (nk)nkk!, for 1kn, from Eq. (13), one can write E(Xr)((ra)(jb+k1))+βkΓ(α)j=0k=0(1)j+k(r/a)jj!((j/b+k1)kk!)Γ(α+k), provided r/aand j/b+k−1 are both integers. Employing successively, the generalized series expansion of (1(1+βu)1/b)j/a, the characteristic function for X~GK(a,b,α,θ)will be given by [from Eq. (3)]

φX(t)=1Γ(α)01eitxf(x)dx=1Γ(α)0uα1euexp(it(1(1+βu)1/b)1/a)duonsubstitutionu=1(1xa)bβ(1xa)b=1Γ(α)j=00(it(1(1+βu)1/b)1/a)jj!uα1eudu=1Γ(α)j=0k1=0k2=0(1)k1+k2βk2(it)j(j/ak1)(k1/bk2)Γ(α+k2).E14

If j/aand k1/bare integers then in Eq. (14), the second and third summations will stop at j/aand k1/b, respectively.

If we denote the median by T, then the mean deviation from the mean, D(μ), and the mean deviation from the median, D(T),can be written as

D(μ)=E|Xμ|=2μG(μ)2μxf(x)dx.E15
D(T)=E|XT|=μ2Txf(x)dx.E16

Now, consider

It=0txf(x)dx=0txabexp(1(1xa)bβ(1xa)b)Γ(α)βα×1(1xa)2(1(1xa)bβ(1xa)b)α1xa1dx.E17

Using the substitution u=1(1xa)bβ(1xa)bin Eq. (17), we obtain

It=1Γ(α)01(1ta)bβ(1ta)b(1(1+uβ)1/b)1/auα1eudu=1Γ(α)j=0k=0(1)jβk(it)j(1/aj)(j/b+k1k)Γ(α,1(1ta)bβ(1ta)b),E18

where we used successively binomial series expansion.

By using Eqs. (2) and (18), the mean deviation from the mean and the mean deviation from the median are, respectively, given by

D(μ)=2μΓ(α,1(1ma)bβ(1ma)b)Γ(α)2Iμ.D(M)=mu2IM.E19

### 4.1. Entropy

One useful measure of diversity for a probability model is given by Renyi’s entropy. It is defined as IR(ρ)=(1ρ)1log(fρ(x)dx), where ρ>0and ρ1. If a random variable Xhas a GK distribution, then we have

fρ(x)=(abΓ(α)βα)ρexp(ρ(1(1xa)b)β(1xa)b)×1(1xa)2ρ(1(1xa)bβ(1xa)b)ρ(α1)xρ(a1)E20

Next, consider the integral

Now, using successive application of the generalized binomial expansion, we can write

(1(1+βu1/ρ)1/b)1/a=(1)ρ1j=0k=0(1)j(ρ1j)(1/b+k1k)βkuk/ρ.E22

Hence, the integral in Eq. (21) reduces to

01fρ(x)dx=(abΓ(α)βα)ρρραk(1)ρ1j=0k=0(1)j(ρ1j)(1/b+k1k)βkΓ(ρα+k)=δ(ρ,α,β,a,b),sayE23

Therefore, the expression for the Renyi’s entropy will be

IR(ρ)=(1ρ)1log(δ(ρ,α,β,a,b))E24

## 5. Maximum likelihood estimation

In this section, we address the parameter estimation of the GK(α, β, a, b) under the classical set up. Let X1, X2, …, Xnbe a random sample of size ndrawn from the density Eq. (3). The log-likelihood function is given by

=αlogβnlogΓ(α)+nloga+nlogb+(a1)i=1nlogXii=1n1(1Xia)bβ(1Xia)b2i=1nlog(1Xia)+(α1)i=1nlog(1(1Xia)bβ(1Xia)b).E25

The derivatives of Eq. (13) with respect to α, β, a, and bare given by

α=nlogβΨ(α)+i=1nlog(1(1Xia)bβ(1Xia)b),E26

where Ψ(α)=αlogΓ(α),

β=αβ+β2i=1n(1(1Xia)bβ(1Xia)b(α1)log(1(1Xia)bβ(1Xia)b)).E27
a=na+i=1nlogXi+2i=1nXia(1Xia)1logXi+b(α1)βi=1n(1(1Xia)bβ(1Xia)b)1XialogXi(1Xia)b+11βi=1nXialogXi(1Xia)b+1E28
b=nb+1β(1+i=1nlog(1Xia)(1(α1β)1(1Xia)b(1Xia)b)).E29

The MLEs α^, β^,a^, and b^are obtained by setting Eqs. (2629) to zero and solving them simultaneously.

To estimate the model parameters, numerical iterative techniques must be used to solve these equations. We may investigate the global maxima of the log likelihood by setting different starting values for the parameters. The information matrix will be required for interval estimation. The elements of the 4 × 4 total observed information matrix (since expected values are difficult to calculate), J(θ)=Jr,s(θ)(for r,s=α,β,a,b), can be obtained from the authors under request, where θ=(α,β,a,b). The asymptotic distribution of (θ^θ)is N4(0,K(θ)1), under the regularity conditions, where K(θ)=E(J(θ))is the expected information matrix, and J(θ^)1is the observed information matrix. The multivariate normal N4(0,K(θ)1)distribution can be used to construct approximate confidence intervals for the individual parameters.

### 5.1. Simulation study

In order to assess the performance of the MLEs, a small simulation study is performed using the statistical software R through the package (stats4), command MLE. The number of Monte Carlo replications was 20,000 For maximizing the log-likelihood function, we use the MaxBFGS subroutine with analytical derivatives. The evaluation of the estimates was performed based on the following quantities for each sample size; the empirical mean squared errors (MSEs) are calculated using the R package from the Monte Carlo replications. The MLEs are determined for each simulated data, say, (α^i,β^i,a^i,b^i)for i=1,2,,20,000, and the biases and MSEs are computed by

biash(n)=120000i=120000(h^ih),E30

and

MSEh(n)=120000i=120000(h^ih)2,E31

for h=α,β,a,b. We consider the sample sizes at n= 100, 200, and 500 and consider different values for the parameters . The empirical results are given in Table 1 . The figures in Table 1 indicate that the estimates are quite stable and, more importantly, are close to the true values for these sample sizes. Furthermore, as the sample size increases, the MSEs decrease as expected.

Sample sizeActual valueBiasMSE
nαβabα^β^a^b^α^β^a^b^
1000.50.524−0.417−0.4190.355−0.3930.0510.0460.0530.97
0.50.535−0.7730.324−0.214−0.3420.0180.0420.0980.626
0.70.8430.489−0.246−0.6230.4820.0150.1210.1060.167
0.90.7640.1880.979−0.5090.0560.0480.0220.0440.114
11.50.90.60.178−0.498−0.429−0.5450.4270.0280.0920.495
1.520.60.8−0.084−0.363−0.405−0.2200.9530.0180.0730.572
2000.50.524.0720.3610.0490.0730.0220.0230.0240.313
0.50.5350.5180.1840.0840.1150.0080.0220.0450.578
0.70.8430.3160.1590.050−0.3290.0060.0590.0440.158
0.90.7640.137−0.0490.131−0.0320.0180.0100.0200.095
11.50.90.60.1250.4750.0860.2420.0640.0130.0280.147
1.520.60.80.0340.1730.224−0.1500.4010.0080.0360.432
5000.50.524−0.046−0.028−0.036−0.0470.0090.0110.0110.084
0.50.535−0.051−0.111−0.0350.0020.0040.0100.0220.018
0.70.843−0.0730−0.0520.046−0.0220.0040.0310.0240.036
0.90.764−0.102−0.02−0.098−0.0230.0080.0050.0100.021
11.50.90.6−0.078−0.052−0.0170.0030.0270.0070.0120.015
1.520.60.80.007−0.0690.066−0.0850.1360.0040.0150.013

### Table 1.

Bias and MSE of the estimates under the maximum likelihood method.

## 6. Reliability parameter

The reliability parameter Ris defined as R=P(X>Y), where Xand Yare independent random variables. For a detailed study on the possible applications of the reliability parameter, an interested reader is suggested to look at Ref. [4, 5]. If Xand Yare two continuous and independent random variables with the cdf’s F1(x)and F2(y)and their pdf’s f1(x)and f2(y), respectively, then the reliability parameter Rcan be written as

R=P(X>Y)=F2(t)f1(t)dt.E32

Theorem 4.Let X~GK(a, b, α1, β1) and Y~(a, b, α2, β2), then

R=p=0(1)pp!(α2+p)Γ(α1)(β1β2)p+α2Γ(α1+α2+p).E33

Proof:From Eqs. (2) and (3), we have

R=01γ1(α2,1(1ta)bβ2(1ta)b)abexp(1(1ta)bβ1(1ta)b)Γ(α1)β1α×1(1ta)2(1(1ta)bβ1(1ta)b)α11ta1dt.E34

Using the series expansion for the incomplete gamma function γ1(k,x)=xkp=0(x)pk!(k+p), and using the substitution u=1(1ta)bβ1(1ta)b, Eq. (34) reduces to

R=p=0(1)pp!(α2+p)Γ(α1)(uβ1β2)p+α2uα11exp(u)du=p=0(1)pp!(α2+p)Γ(α1)(β1β2)p+α2Γ(α1+α2+p).E35

Hence the proof. W

## 7. Order statistics

Here, we derive the general r-th order statistic and the large sample distribution of the sample minimum and the sample maximum based on a random sample of size nfrom the GK(α, β, a, b) distribution. The corresponding density function of the r-th order statistic, Xr:n,from Eq. (3) will be

fXr:n(x)=1B(r,nr+1)(F(x))r1(1F(x))nrf(x)=f(x)B(r,nr+1)j=0r1(1)j(r1j)(Γ(α,β11(1xr:na)b(1xr:na)b)Γ(α))nr+j×I(0<x<1).E36

Using the series expression for the incomplete gamma function: γ1(α,x)=k=0ex(x)α+kα(α+1)(α+k), the pdf of Xr:ncan be written as

fr:n(x)=1B(r,nr+1)f(x)j=0r1(1)j(r1j)(k=0exp(1(1xr:na)bβ(1xr:na)b)(1(1xr:na)bβ(1xr:na)b)α+kΓ(α)α(α+1)(α+k))nr+j=f(x)B(r,nr+1)j=0r1k1knr+j=0(1)j+sk(r1j)exp((nr+j)1(1xr:na)bβ(1xr:na)b)×(1(1xr:na)b(1xr:na)b)sk+(nr+j)α(Γ(α))nr+jβsk+(nr+j)αpk=1B(r,nr+1)j=0nrk1knr+j=0(1)j+sk(r1j)×Γ(sk+(r+j)α)(Γ(α))nr+jpkf(x|sk+(nr+j)α,β,a,b),E37

where sk=i=1nr+jkiand pk=i=1nr+j(ki+α).

From Eq. (37), it is interesting to note that the pdf of the r-th order statistic Xr:ncan be expressed as an infinite sum of the GK pdf ’s.

## 8. Application

Here, we consider two well-known illustrative data sets which are used to show the efficacy of the GK distribution. For details on these two data sets [6, 7], the second data set in Table 2 is from Ref. [8], and it represents the fatigue life of 6061-T6 aluminum coupons cut parallel with the direction of rolling and oscillated at 18 cycles per second. The GK distribution is fitted to the first data set and compared the result with the Kumaraswamy, gamma-uniform [9], and beta-Pareto [10]. These results are reported in Table 3 . The results show that gamma-uniform, GK distributions provide adequate fit to the data. Figure 3 displays the empirical and the fitted cumulative distribution functions. This figure supports the results in Table 3 . A close look at Figure 3 indicates that GK distribution provides better fit to the left tail than the gamma-uniform distribution. This is due to the fact that GK distribution can have longer left tail ( Figure 3 ).

7090969799100103104104105
107108108108109109112112113114
114114116119120120120121121123
124124124124124128128129129130
130130131131131131131132132132
133134134134134134136136137138
142142144144145146148148149151
151152155156157157157157158159
162163163164166166168170174196
212

### Table 2.

Fatigue life of 6061-T6 aluminum data.

DistributionKumaraswamyGamma-uniformBeta-ParetoGamma-Kumaraswamy
Parameter estimatesa^=0.653α^=7.528c^=5.048α^=7.891
b^=1.1182β^=2.731β^=0.401β^=0.785
a^=6.49θ^=6.417a^=5.352
b^=0.932b^=1.735
Log likelihood−162.34−116.58−113.36−113.25
AIC217.38119.45167.78107.85
AICC218.36120.37168.91108.43
CAIC218.36120.37168.91108.43
A0*12.1640.59511.31250.4282
W0*2.89370.09310.13170.04893
K-S0.52900.095210.42450.0492
K-S p-value0.00000.91400.83740.9978

### Table 3.

Goodness of fit of deep-groove ball bearings data.

In addition, to check the goodness-of-fit of all statistical models, several other goodness-of-fit statistics are used and are computed using computational package Mathematica. The MLEs are computed using N maximize technique as well as the measures of goodness-of-fit statistics including the log-likelihood function evaluated at the MLEs (l), Akaike information criterion (AIC), corrected Akaike information criterion (AICC), consistent Akaike information criterion (CAIC), the Anderson-Darling (A*), the Cramer-von Mises (W*), and the Kolmogrov-Smirnov (K-S) statistics with their p values to compare the fitted models. These statistics are used to evaluate how closely a specific distribution with cdf (2) fits the corresponding empirical distribution for a given data set. The distribution with better fit than the others will be the one having the smallest statistics and largest p value. Alternatively, the distribution for which one obtains the smallest of each of these criteria (i.e., AIC, AICC, K-S, etc.) will be most suitable one. The mathematical equations of those statistics are given by

• AIC=2(θ^)+2q

• AICC=AIC+2q(q+1)nq1

• CAIC=2(θ^)+2qnnq1

• A0*=(2.25n2+0.75n+1)(n1ni=1n(2i1)log(zi(1zni+1)))

• W0*=(0.5n+1)[(zi2i12n)2+112n]

• KS=Max(inzi,zii1n),

where (θ^)denotes the log-likelihood function evaluated at the maximum likelihood estimates, qis the number of parameters, nis the sample size and zi=cdf(yi), the yi’s being the ordered observations.

Lieblein and Zelen [6] proposed a five parameter beta generalized Pareto distribution and fitted the data in Table 4 and compared the result with beta-Pareto and other known distributions. The results of fitting beta generalized Pareto and beta-Pareto from Ref. [8] are reported in Table 4 along with the results of fitting the Pareto (IV) and GK distributions to the data. The KS value from Table 4 indicates that the GK distribution provides the best fit. The fact that GK distribution has the least number of parameters than beta generalized Pareto and beta-Pareto adds an extra advantage over them. Figure 4 displays the empirical and the fitted cumulative distribution functions. This figure supports the results in Table 4 .

DistributionKumaraswamyBeta-ParetoBeta generalized Paretogamma-Kumaraswamy
Parameter estimatesδ^=0.235α^=485.470α^=12.112α^=4.87
γ^=2.4926β^=162.060β^=1.702β^=3.352
k^=0.3943μ^=40.564a^=1.1722
θ^=3.910k^=0.273b^=2.0154
θ^=54.837
Log likelihood−572.39−517.33−457.85−417.36
AIC1018.83925.30925.70878.45
AICC1018.956926.01926.84879.63
CAIC1018.956926.01926.84879.63
A0*4.17451.00830.65840.4921
W0*0.67390.28270.11950.0822
K-S0.47230.0910.0700.064
K-S p-value0.0000.2480.5370.736

### Table 4.

Parameter estimates for the fatigue life of 6061-T6 aluminum coupons data.

## 9. Characterization of GK distribution

In this section, we present characterizations of GK distribution in terms of the ratio of two truncated moments. For the previous works done in this direction, we refer the interested readers to Glänzel [1114] and Hamedani [1517]. For our characterization results, we employ a theorem due to Ref. [11], see for further details. The advantage of the characterizations given here is that cdfFneed not have a closed form. We present here a corollary as a direct application of the theorem discussed in details in Ref. [11].

Corollary 1.Let X:Ω(0,1)be a continuous random variable and let h(x)=βα1(1xa)b(α2)+1[1(1xa)b]1αand g(x)=h(x)exp(1(1xa)bβ(1xa)b)for x(0,1).Then Xhas pdf (3) if and only if the function ηdefined in Theorem 5 has the form

η(x)=12exp(1(1xa)bβ(1xa)b),0<x<1.E38

Proof. Let Xhas pdf (3), then

(1F(x))E[h(X)|Xx]=1βα1Γ(α)exp(1(1xa)bβ(1xa)b),0<x<1,E39

and

(1F(x))E[g(X)|Xx]=12βα1Γ(α)exp{2(1(1xa)bβ(1xa)b)},0<x<1,E40

and finally

η(x)h(x)g(x)=h(x)2exp(1(1xa)bβ(1xa)b)<0,for0<x<1.E41

Conversely, if ηis given as above, then

s(x)=η(x)h(x)η(x)h(x)g(x)=abβxa1(1xa)(b+1),0<x<1,E42

and hence

s(x)=1β(1xa)b,0<x<1.E43

Now, in view of Theorem 5, Xhas pdf (3).

Corollary 2.Let X:Ω(0,1)be a continuous random variable and let h(x) be as in Proposition 1. Then, Xhas pdf (3) if and only if there exist functions gand ηdefined in Theorem 5 satisfying the differential equation

η(x)h(x)η(x)h(x)g(x)=abβxa1(1xa)(b+1),0<x<1.E44

Remarks 1.(a) The general solution of the differential equation in Corollary 1 is

η(x)=exp(1(1xa)bβ(1xa)b)[abβxa1(1xa)(b+1)exp(1(1xa)bβ(1xa)b)(h(x))1g(x)dx+D],E45

for 0 < x< 1, where Dis a constant. One set of appropriate functions is given in Proposition 1 with D= 0

(b) Clearly, there are other triplets of functions (h, g, η) satisfying the conditions of Theorem 5. We presented one such triplet in Proposition 1.

## 10. Concluding remarks

A special case of the gamma-generated family of distributions, the gamma-Kumaraswamy distribution, is defined and studied. Various properties of the gamma-Kumaraswamy distribution are investigated, including moments, hazard function, and reliability parameter. The new model includes as special sub-models the gamma and Kumaraswamy distribution. Also, we provide various characterizations of the gamma-Kumaraswamy distribution. An application to a real data set shows that the fit of the new model is superior to the fits of its main sub-models. As future work related to this univariate GK model, we will consider the following:

• A natural bivariate extension to the model in Eq. (1) would be

f(x,y)g(x,y)Γ(α)βαG¯2(x,y)exp(g(x,y)βbarG(x,y))(g(x,y)βbarG(x,y))α1,x>0,y>0.E46

In this case, exact evaluation of the normalizing constant would be difficult to obtain, even for a simple analytic expression of a baseline bivariate distribution function, G(x, y). Numerical methods such as Monte Carlo methods of integration might be useful here. We will study and discuss structural properties of such a bivariate GK model.

• Extension of the proposed univariate GK model to multivariate GK models and discuss the associated inferential issues. It is noteworthy to mention that classical methods of estimation, such as for example, maximum likelihood method of estimation might not be a good strategy because of the enormous number of model parameters. An appropriate Bayesian inference might be the only remedy. In that case, we will separately study two different cases of estimation: (a) with non-informative priors and (b) with full conditional conjugate priors (Gibbs sampling). Since the GK distribution is in the one parameter exponential family, a reasonable choice for priors for αand βmight well be gamma priors with appropriate choice of hyper-parameters. For prior choices of the parameters that are from the baseline G(.) distribution function, a data-driven prior approach will be more suitable.

• A discrete analog of the univariate GK model with a possible application in modeling rare events.

• Construction of a new class of GK mixture models by adopting Marshall-Olkin method of obtaining new distribution.

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Written By

Indranil Ghosh and Gholamhossein G. Hamedani

Submitted: April 14th, 2016 Reviewed: November 9th, 2016 Published: April 26th, 2017