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The Kumaraswamy-Generalized Gamma Distribution
with Application in Survival Analysis
Marcelino A. R. PascoaDepartment of Exact Sciences, ESALQ - USP
Edwin M. M. Ortega1
Department of Exact Sciences, ESALQ - USPGauss M. Cordeiro
Department of Statistics and Informatics, DEINFO - UFRPEPatrıcia F. Paranaıba
Department of Exact Sciences, ESALQ - USP
Abstract
Based on the Kumaraswamy distribution (Jones, 2009), we study the so-called Kum-generali-zed gamma distribution that is capable of modeling bathtub-shaped hazard rate functions. Thebeauty and importance of this distribution lies in its ability to model monotone and non-monotone failure rate functions, which are quite common in lifetime data analysis and relia-bility. The new distribution has a number of well-known lifetime special sub-models such asthe exponentiated generalized gamma, exponentiated Weibull, exponentiated generalized half-normal, exponentiated gamma, generalized Rayleigh, among several others. Some mathematicalproperties of the Kum-generalized gamma distribution are studied. We obtain two infinite sumrepresentations for the moments and for the moment generating function. We calculate thedensity of the order statistics and two expansions for their moments. The method of maximumlikelihood is used for estimating the model parameters and the observed information matrix isderived. Two real data sets are analyzed with this distribution.Keywords: Censored data; Data analysis; Generalized gamma distribution; Maximum likelihoodestimation; Moment; Order statistic.
1 Introduction
The gamma distribution is the most popular model for analyzing skewed data. The gene-ralized gamma distribution (GG) was introduced by Stacy (1962) and includes as special sub-models: the exponential, Weibull, gamma and Rayleigh distributions, among others. The GG
1Address for correspondence: Departamento de Ciencias Exatas, ESALQ/USP, Av. Padua Dias 11 - Caixa Postal9, 13418-900, Piracicaba - Sao Paulo - Brazil. e-mail: [email protected]
1
distribution is suitable for modeling data with different types of hazard rate function: increasing,decreasing, in the form of bathtub and unimodal. This characteristic is useful for estimatingindividual hazard rate functions and both relative hazards and relative times (Cox, 2008). Recently,the GG distribution has been used in several research areas such as engineering, hydrology andsurvival anlaysis. Ortega et al. (2003) discussed influence diagnostics in GG regression models,Nadarajah and Gupta (2007) used this distribution with application to drought data, Cox et al.(2007) presented a parametric survival analysis and taxonomy of the GG hazard rate functions andAli et al. (2008) derived the exact distributions of the product X1X2 and the quotient X1/X2, whenX1 and X2 are independent GG random variables providing applications of their results to droughtdata from Nebraska. Further, Gomes et al. (2008) focused on the parameter estimation, Ortegaet al. (2008) compared three type of residuals based on the deviance component in GG regressionmodels under censored observations, Cox (2008) discussed and compared the F-generalized familywith the GG model, Almpanidis and Kotropoulos (2008) presented a text-independent automaticphone segmentation algorithm based on the GG distribution and Nadarajah (2008a) analyzedsome incorrect references with respect to the use of this distribution in electrical and electronicengineering. More recently, Barkauskas et al. (2009) modeled the noise part of a spectrum asan autoregressive moving average (ARMA) model with innovations having the GG distribution,Malhotra et al. (2009) provided a unified analysis for wireless system over generalized fadingchannels that is modeled by a two parameter GG model and Xie and Liu (2009) analyzed three-moment auto conversion parametrization based on the GG distribution. Further, Ortega et al.(2009) proposed a modified GG regression model to allow the possibility that long-term survivorsmay be presented in the data and Cordeiro et al. (2009) proposed the exponentiated generalizedgamma (EGG) distribution.
In the last decade, several authors have proposed new classes of distributions, which are basedon modifications (in different ways) of the Weibull distribution to provide hazard rate functionshaving the form of U . Among these, we mention the Weibull, exponentiated Weibull (Mudholkar etal., 1995), which also shows unimodal hazard rate function, the additive Weibull (Xie and Lai, 1995)and the extended Weibull (Xie et al., 2002) distributions. Recently, Carrasco et al. (2008) presenteda four-parameter model so-called the generalized modified Weibull (GMW) distribution, Gusmaoet al. (2009) introduced and studied a three-parameter generalized inverse Weibull distributionwith decreasing and unimodal failure rate and Pescim et al. (2009) proposed a four-parameterdistribution, so-called the beta generalized half normal distribution.
The Kumaraswamy distribution (Kumaraswamy, 1980) is not very common among statisticiansand has been little explored in the literature. We refer to the Kum distribution to denote theKumaraswamy distribution. Its cumulative distribution function (cdf) is defined by F (x) = 1 −(1 − xλ)ϕ, 0 < x < 1, where λ > 0 and ϕ > 0 are two additional parameters whose role is tointroduce asymmetry and produce distributions with heavier tails. The Kum distribution doesnot seem to be very familiar to statisticians and has not been investigated systematically in muchdetail before, nor has its relative interchangeability with the beta distribution has been widelyappreciated.
2
However, in a very recent paper, Jones (2009) explored the background and genesis of the Kumdistribution and, more importantly, made clear some similarities and differences between the betaand Kum distributions. He highlighted several advantages of the Kum distribution over the betadistribution: the normalizing constant is very simple; simple explicit formulae for the distributionand quantile functions which do not involve any special functions; a simple formula for randomvariate generation; explicit formulae for L-moments and simpler formulae for moments of orderstatistics. Further, according to Jones (2009), the beta distribution has the following advantagesover the Kum distribution: simpler formulae for moments and moment generating function (mgf);a one-parameter sub-family of symmetric distributions; simpler moment estimation and more waysof generating the distribution via physical processes.
The probability density function (pdf) of the Kum distribution also has a simple form f(x) =λϕxλ−1(1 − xλ)ϕ−1, and it can be unimodal, increasing, decreasing or constant, depending in thesame way on the values of its parameters like the beta distribution.
If G(x) is the baseline cdf of a random variable, the cdf of the Kum-generalized distribution,say Kum-G distribution, is defined by (Cordeiro and Castro, 2010)
F (x) = 1− [1−G(x)λ]ϕ, (1)
where λ > 0 and ϕ > 0 are two additional parameters whose role is to introduce asymmetry andgenerate a distribution with heavier tails. The density function corresponding to (1) is
f(x) = λϕg(x)G(x)λ−1[1−G(x)λ]ϕ−1, (2)
where g(x) = dG(x)/dx. The density (2) does not involve any special function, such as theincomplete beta function, as was established in the class of generalized beta distributions proposedby Eugene et al. (2002). So, the new Kum-G distribution is obtained by adding two parametersλ and ϕ to the quantile function of the G distribution. This generalization contains distributionswith unimodal and bathtub shaped hazard rate functions. It also contemplates a broad class ofmodels with monotone risk functions. Some mathematical properties of the Kum-G distributionderived by Cordeiro and Castro (2010) are usually much simpler than those properties of the betaG distribution (Eugene et al., 2002).
In this article, we combine the works of Kumaraswamy (1980), Cordeiro et al. (2009) andCordeiro and Castro (2010) to study the mathematical properties of the so-called Kum-generalizedgamma (KumGG) distribution. The rest of the paper is organized as follows. Section 2 introducesthe KumGG distribution. Some important special models are presented in Section 3. A range ofmathematical properties of the KumGG distribution is considered in Sections 4 to 9. The KumGGdensity is straightforward to compute using any statistical software with numerical facilities. How-ever, in Section 4, we demonstrate that the KumGG distribution can be written as a mixture ofGG distributions. Section 5 provides two explicit expansions for the moments. In Section 6, twoclosed form expressions for the moment generating function (mgf) are derived. Mean deviationsand Bonferroni and Lorenz curves are obtained in Sections 7 and 8, respectively. In Section 9,
3
Table 1: Some GG Distributions.
Distribution τ α k
Gamma 1 α kChi-square 1 2 n
2
Exponential 1 α 1Weibull c α 1Rayleigh 2 α 1Maxwell 2 α 3
2
Folded normal 2√
2 12
we demonstrate that the density function of the KumGG order statistics is a mixture of KumGGdensities. As an application of this result, we also provide expansions for the moments of the orderstatistics. Maximum likelihood estimation of the model parameters is discussed in Section 10. Tworeal lifetime data sets are used in Section 11 to illustrate the usefulness of the KumGG model.Concluding remarks are given in Section 12.
2 The Kumaraswamy-Generalized Gamma Distribution
Let G(t; α, τ, k) be the cdf of the GG distribution (Stacy, 1962) given by
G(t;α, τ, k) =γ(k, (t/α)τ )
Γ(k),
where α > 0, τ > 0, k > 0, γ(k, x) =∫ x0 wk−1e−wdw is the incomplete gamma function and Γ(.)
is the gamma function. Basic properties of the GG distribution are given by Stacy and Mihram(1965) and Lawless (1980, 2003). Some important distributions that are special sub-models of theGG distribution are listed in Table 1.
The cdf of the KumGG distribution can be defined by substituting G(t; α, τ, k) into equation(1). Hence, the density function of the KumGG distribution with five parameters α > 0, τ > 0,k > 0, λ > 0 and ϕ > 0 has the form (for t > 0)
f(t) =λϕτ
αΓ(k)
(t
α
)τk−1
exp
[−
(t
α
)τ]{
γ1
[k,
(t
α
)τ]}λ−1(
1−{
γ1
[k,
(t
α
)τ]}λ)ϕ−1
. (3)
Here, γ1(·, ·) is the incomplete gamma ratio function defined by γ1(k, x) = γ(k, x)/Γ(k), i.e. the cdfof the standard gamma distribution with parameter k. In the density (3), α is a scale parameterand the other parameters τ , k and λ are shape parameters. It can be immediately verified thatequation (3) is a density function.
4
The Weibull and GG distributions are clearly the most important sub-models of (3) for ϕ =λ = k = 1 and ϕ = λ = 1, respectively. The KumGG distribution approaches the log-normal(LN) distribution when ϕ = λ = 1 and k → ∞. Other sub-models can be immediately definedfrom Table 1: Kum-Gamma, Kum-Chi-Square, Kum-Exponential, Kum-Weibull, Kum-Rayleigh,Kum-Maxwell and Kum-Folded normal with 4, 3, 3, 4, 3, 3 and 2 parameters, respectively.
If T is a random variable with density function (3), we write T ∼ KumGG(α, τ, k, λ, ϕ). Thesurvival and hazard rate functions corresponding to (3) are
S(t) = 1− F (t) =
(1−
{γ1
[k,
(t
α
)τ]}λ)ϕ
(4)
and
h(t) =λϕτ
αΓ(k)
(t
α
)τk−1
exp
[−
(t
α
)τ]{
γ1
[k,
(t
α
)τ]}λ−1 (
1−{
γ1
[k,
(t
α
)τ]}λ)−1
, (5)
respectively. Plots of the KumGG density and survival rate function for selected parameter valuesare given in Figures 1 and 2, respectively.
If T ∼ KumGG(α, τ, k, λ, ϕ), then the survival function S(t) of T can be expressed as
S(t) =
{1−
[1
ατkΓ(k)
∞∑
j=0
(−1)jα−(τj)tτ(k+j)
(k + j)j!
]λ}ϕ
. (6)
We simulate the KumGG distribution by solving the nonlinear equation(1− u1/ϕ
)1/λ − γ1
[k,
( t
α
)τ]= 0, (7)
where u has the uniform U(0, 1) distribution.Some properties of the KumGG distribution are given below:
If T ∼ KumGG(α, τ, k, λ, ϕ) ⇒ bT ∼ KumGG(bα, τ, k, λ, ϕ), ∀b > 0.
If X ∼ KumGG(1, 1, k, λ, ϕ) ⇒ T = αX1/τ ∼ KumGG(α, τ, k, λ, ϕ).
The latter result is important because it provides a simple way to generate random variablesfollowing the KumGG distribution.
An important property of the KumGG distribution for information analysis is that this distri-bution is closed under power transformation:
If T ∼ KumGG(α, τ, k, λ, ϕ) ⇒ Tm ∼ KumGG(αm, τ/m, k, λ, ϕ), ∀m 6= 0.
The KmGG distribution has a hazard rate function that involves a complicated function, al-though it can be easily computed numerically. Moreover, it is quite flexible for modeling survivaldata. Plots of the KumGG hazard rate shapes for selected parameter values are given in Figure 3.
5
(a) (b)
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
t
f(t)
α=1; τ=1; k=1; λ=2;ϕ=2.5α=1.1; τ=1.3; k=0.9; λ=3; ϕ=0.8α=1; τ=2; k=1; λ=3; ϕ=0.7α=1; τ=1; k=1; λ=0.5; ϕ=1α=1; τ=1; k=1; λ=2.5; ϕ=0.5
2.0 2.5 3.0 3.5 4.0 4.5 5.00.
00.
51.
01.
52.
0
t
f(t)
α=1; τ=1; k=1; λ=120;ϕ=15α=0.9; τ=1; k=0.8; λ=100; ϕ=3α=1; τ=1.2; k=1; λ=134; ϕ=10α=1; τ=1; k=0.7; λ=130; ϕ=3α=1; τ=1; k=1.5; λ=150; ϕ=90
(c)
0 1 2 3 4 5
0.0
0.5
1.0
1.5
t
f(t)
α=1; τ=1; k=1; λ=2;ϕ=2.5α=1.1; τ=1.3; k=0.9; λ=3; ϕ=0.8α=1; τ=1; k=1.5; λ=150; ϕ=90α=1; τ=1; k=1; λ=0.5; ϕ=1α=1; τ=1; k=0.7; λ=130; ϕ=3α=1; τ=1; k=1; λ=120; ϕ=15
Figure 1: Plots of the KumGG density for selected parameter values.
3 Special Distributions
The following well-known and new distributions are special sub-models of the KumGG distri-bution.
6
(a) (b)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
t
S(t
)
α=1; τ=1; k=1; λ=0.5;ϕ=1α=1; τ=1; k=1; λ=10; ϕ=1α=1; τ=1; k=1; λ=30; ϕ=1α=1; τ=1; k=1; λ=60; ϕ=1α=1; τ=1; k=1; λ=100; ϕ=1
0 2 4 6 8 100.
00.
20.
40.
60.
81.
0
t
S(t
)
α=1; τ=1; k=1; λ=1;ϕ=0.5α=1; τ=1; k=1; λ=1; ϕ=1α=1; τ=1; k=1; λ=1; ϕ=1.5α=1; τ=1; k=1; λ=1; ϕ=2α=1; τ=1; k=1; λ=1; ϕ=2.5
Figure 2: (a) Plots of the KumGG survival function for some values of λ. (b) Plots of the KumGGsurvival function for some values of ϕ.
• Exponentiated Generalized Gamma distribution
If ϕ = 1, the KumGG distribution reduces to
f(t) =λτ
αΓ(k)
(t
α
)τk−1
exp
[−
(t
α
)τ]{
γ1
[k,
(t
α
)τ]}λ−1
, t > 0,
which is the exponentiated generalized gamma (EGG) density introduced by Cordeiro et al.(2009). If τ = ϕ = 1 in addition to k = 1, the special case corresponds to the exponentiatedexponential (EE) distribution proposed by Gupta and Kundu (1999, 2001). If τ = 2 inaddition to k = ϕ = 1, the special case becomes the generalized Rayleigh (GR) distribution(Kundu and Raqab, 2005).
• Kum-Weibull distribution (Cordeiro and Castro, 2010)
7
(a) (b)
0 2 4 6 8 10
01
23
45
6
t
h(t)
α=2.3; τ=2; k=0.4; λ=0.1;ϕ=1.3α=2.3; τ=2; k=1; λ=0.1; ϕ=1.3α=2.3; τ=2; k=2; λ=0.1; ϕ=2α=2.1; τ=2; k=2.1; λ=0.1; ϕ=2α=2; τ=2; =2; λ=0.1; ϕ=2
0 2 4 6 8 100
12
34
56
7
t
h(t)
α=2; τ=2; k=2; λ=0.5;ϕ=3α=2; τ=3; k=2; λ=0.5; ϕ=1α=1.8; τ=2; k=2.2; λ=0.7; ϕ=5α=1; τ=1; k=3; λ=1; ϕ=10α=1.5; τ=4; k=2.3; λ=0.6; ϕ=4
(c)
0 2 4 6 8 10
01
23
4
t
h(t)
α=2; τ=1; k=2; λ=0.01;ϕ=2.4α=2.1; τ=1; k=2; λ=0.01; ϕ=3α=3; τ=2.2; k=4.1; λ=1.3; ϕ=1.5α=3; τ=2.1; k=4; λ=1.5; ϕ=1.3α=1; τ=1; k=1; λ=1; ϕ=1
Figure 3: KumGG hazard rate functions. (a) The distribution has a bathtub hazard rate function.(b) The distribution has an unimodal hazard rate function. (c) The distribution has increasing,decreasing and constant hazard rate function.
8
For k = 1, equation (3) yields
f(t) =τλϕ
α
(t
α
)τ−1
exp
[−
(t
α
)τ]{
1− exp
[−
(t
α
)τ]}λ−1
×(
1−{
1− exp
[−
(t
α
)τ]}λ)ϕ−1
, t > 0,
which is the Kum-Weibull (KumW) distribution. If ϕ = k = 1, it reduces to the exponentiatedWeibull (EW) distribution (see, Mudholkar et al., 1995, 1996). If ϕ = λ = k = 1, (3)becomes the Weibull distribution. If τ = 2 and k = 1, we obtain the Kum-Rayleigh (KumR)distribution. If k = τ = 1, we obtain the Kum-exponential (KumE) distribution. If ϕ = λ =k = 1, we obtain two important special sub-models: the exponential (τ = 1) and Rayleigh(τ = 2) distributions, respectively.
• Kum-Gamma distribution (Cordeiro and Castro, 2010)
For τ = 1, the KumGG distribution reduces to
f(t) =λϕ
αΓ(k)
(t
α
)k−1
exp
[−
(t
α
) ]{γ1
[k,
(t
α
)]}λ−1
×(
1−{
γ1
[k,
(t
α
)]}λ)ϕ−1
, t > 0,
which is the four parameter Kum-Gamma (KumG4) distribution. If ϕ = τ = 1, we obtainthe exponentiated gamma (EG3) distribution with three parameters. If ϕ = τ = α = 1, thespecial case corresponds to the exponentiated gamma (EG2) distribution with two parameters.Further, if k = 1, we obtain the Kum-Gamma distribution with one parameter. If we takeϕ = λ = τ = 1, the special case corresponds to the two parameter gamma distribution. Inaddition, if k = 1, we obtain the one parameter gamma distribution.
• Kum-Chi-Square distribution (new)
If we take τ = 2, α = 2 and k = p2 , the density is given by
f(t) =λϕ
2Γ(p2)
(t
2
) p2−1
exp
[−
(t
2
) ][γ1
(p
2,t
2
)]λ−1{1−
[γ1
(p
2,t
2
)]λ}ϕ−1
, t > 0,
which is the Kum-Chi-Square (Kum-Chi) distribution. If ϕ = 1, α = τ = 2 and k = p2 ,
we obtain the exponentiated-chi-square (E-Chi) distribution. If ϕ = λ = 1, in addition toα = τ = 2 and k = p/2, we obtain the well-known chi-square distribution.
9
• Kum-Scaled Chi-Square distribution (new)
For τ = 1, α =√
2σ and k = p2 , the KumGG distribution becomes
f(t) =2λϕ√2σΓ(p
2)
(t√2σ
)p−1
exp
[−
(t2
2σ2
) ]{γ1
[p
2,
(t√2σ
)2]}λ−1
×(
1−{
γ1
[p
2,
(t√2σ
)2]}λ)ϕ−1
, t > 0,
which is the Kum-Scaled Chi-Square (KumSChi) distribution. For ϕ = τ = 1, α =√
2σ andk = p
2 , we obtain the exponentiated scaled chi-square (ESChi) distribution. If ϕ = λ = 1, inaddition to α =
√2σ, τ = 1 and k = p/2, the special case coincides with the scaled chi-square
(SChi) distribution.
• Kum-Maxwell distribution (new)
For τ = 2, α =√
θ and k = 32 , the KumGG distribution reduces to
f(t) =4λϕ√πθ3/2
t2 exp
[−
(t2
θ
)][γ1
(32,t2
θ
)]λ−1{1−
[γ1
(32,t2
θ
)]λ}ϕ−1
, t > 0,
which is the Kum-Maxwell (KumMa) distribution. For ϕ = 1, τ = 2, α =√
θ and k = 32 , we
obtain the exponentiated Maxwell (EM) distribution. If ϕ = λ = 1 in addition to α =√
θ,τ = 2 and k = 3/2, it reduces to the Maxwell (Ma) distribution (see, for example, Bekkerand Roux, 2005).
• Kum-Nakagami distribution (new)
For τ = 2, α =√
w/µ and k = µ, the KumGG distribution becomes
f(t) =2µµλϕ
wµΓ(µ)t2µ−1 exp
[−
(µt2
w
) ][γ1
(µ,
µt2
w
)]λ−1{1−
[γ1
(µ,
µt2
w
)]λ}ϕ−1
, t > 0,
which is the Kum-Nakagami (KumNa) distribution. For ϕ = 1, τ = 2, α =√
w/µ and k = µ,we obtain the exponentiated Nakagami (EM) distribution. If ϕ = λ = 1, in addition toα =
√w/µ, τ = 2 and k = µ, the special case corresponds to the Nakagami (Na) distribution
(see, for example, Shankar et al., 2001).
• Kum-Generalized Half-Normal distribution (new)
10
If τ = 2γ, α = 212γ θ and k = 1
2 , the KumGG distribution becomes
f(t) =λϕγ
t
√2π
(t
θ
)γ
exp
[−1
2
(t
θ
)2γ]{
γ1
[12,12
(t
θ
)2γ]}λ−1
×(
1−{
γ1
[12,12
(t
θ
)2γ]}λ)ϕ−1
, t > 0,
which is referred to as the Kum-generalized half-normal (KumGHN) distribution. For ϕ = 1,τ = 2γ, α = 2
12γ θ and k = 1/2, we obtain the exponentiated generalized half-normal (EGHN)
distribution. For α = 212 θ, τ = k = 2, we obtain the Kum-half normal (KumHN) distribution.
If ϕ = 1, α = 212 θ, τ = 2 and k = 1/2, the reduced model is called the exponentiated half-
normal (EHN) distribution. If ϕ = λ = 1, in addition to α = 212γ θ, τ = 2γ, k = 1/2,
the reduced model becomes the generalized half-normal (GHN) distribution introduced byCooray and Ananda (2008). Further, if ϕ = λ = 1 in addition to α = 2
12 θ, τ = 2 and k = 1/2,
it reduces to the well-known half-normal (HN) distribution.
In appendix B, Table 4 provides a summary of these sub-models.
4 Expansion for the Density Function
Let gα,τ,k(t) be the density function of the GG(α, τ, k) distribution given by
gα,τ,k(t) =τ
αΓ(k)
(t
α
)τk−1
exp
[−
(t
α
)τ], t > 0.
From equation (3) and using the expansion
(1− z)b−1 =∞∑
j=0
(−1)jΓ(b)Γ(b− j)j!
zj ,
which holds for |z| < 1 and b > 0 real non-integer, the density function of the KumGG(α, τ, k, λ, ϕ)distribution can be written as
f(t) =λϕτ
αΓ(k)
(t
α
)τk−1
exp
[−
(t
α
)τ]{
γ1
[k;
(t
α
)τ]}λ−1 ∞∑
j=0
(−1)jΓ(ϕ)Γ(ϕ− j)j!
{γ1
[k,
( t
α
)τ]}λj
. (8)
Using the expansion (32) given in Appendix A in (8), we obtain
11
f(t) =λϕτ
αΓ(k)
( t
α
)τk−1exp
[−
(t
α
)τ] ∞∑
m,j=0
(−1)jΓ(ϕ)sm(λ)Γ(ϕ− j)j!
{γ1
[k,
( t
α
)τ]}λj+m
. (9)
If λj + m > 0 is real non-integer, we have{
γ1
[k,
( t
α
)τ]}λj+m
=(
1−{
1− γ1
[k,
( t
α
)τ]})λj+m+1−1
=∞∑
l=0
(−1)l
(λj + m
l
){1− γ1
[k,
( t
α
)τ]}l
.
Using the binomial expansion in the above expression, we can rewrite (9) as
f(t) =λϕτ
αΓ(k)
( t
α
)τk−1exp
[−
(t
α
)τ] ∞∑
j,l,m=0
l∑
q=0
(−1)j+l+qΓ(ϕ)sm(λ)Γ(ϕ− j)j!
×(
λj + m
l
)(l
q
) {γ1
[k,
( t
α
)τ]}q
. (10)
Using the expansion (37) given in Appendix A, we obtain the mixture density
f(t) =∞∑
d,j,l,m=0
l∑
q=0
w(d, j, l, q, k,m, λ, ϕ)g(α,τ,k(1+q)+d)(t), t > 0, (11)
whose weighted coefficients are given by
w(d, j, l, q, k, m, λ, ϕ) =(−1)j+l+qλϕΓ(ϕ)Γ[k(1 + q) + d]sm(λ)cq,d
Γ(k)q+1Γ(ϕ− j)j!
(λj + m
l
)(l
q
).
Here, the coefficients satisfy∑∞
m,i=0 w(d, j, l, q, k, m, λ, ϕ) = 1, the quantity sm(λ) is given by(33) and the coefficients cq,d can be determined from the recurrence relation (36). Equation (11)shows that the KumGG density can be written in terms of a mixture of GG densities. Hence, somemathematical properties of the KumGG distribution can follow directly from those properties of theGG distribution. For example, the ordinary, inverse and factorial moments, mgf and characteristicfunction of the KumGG distribution can be obtained directly from the GG distribution.
5 Moments
We hardly need to emphasize the necessity and importance of moments in any statistical analysisespecially in applied work. Some of the most important features and characteristics of a distribution
12
can be studied through moments (e.g., tendency, dispersion, skewness and kurtosis). In this section,we give two different expansions for determining the moments of the KumGG distribution. Letµ′r = E(T r) be the rth ordinary moment of the KumGG distribution.
First, we obtain an infinite sum representation for µ′r from equation (11). The rth moment ofthe GG(α, β, k) distribution is
µ′r,GG =αrΓ(k + r/β)
Γ(k).
Equation (11) then immediately gives
µ′r = αr∞∑
d,j,l,m=0
l∑
q=0
(−1)j+l+qλϕΓ(ϕ)Γ[k(1 + q) + d + r/τ ]sm(λ)cq,d
Γ(k)q+1Γ(ϕ− j)j!
(λj + m
l
)(l
q
). (12)
Equation (12) has the inconvenient that the moment µ′r depends on the quantities cq,d that canonly be calculated recursively from equation (36).
We now derive another infinite sum representation for µ′r by computing it directly
µ′r =λϕταr−1
Γ(k)
∫ +∞
0
(t
α
)τk+r−1
exp
[−
(t
α
)τ]{
γ1
[k,
(t
α
)τ]}λ−1
×(
1−{
γ1
[k,
(t
α
)τ]}λ)ϕ−1
dt. (13)
Setting x =(
tα
)τ in (13) yields
µ′r =λϕαr
Γ(k)
∫ +∞
0xk+ r
τ−1 exp (−x) γ1 (k, x)λ−1
[1− γ1 (k, x)λ
]ϕ−1dx. (14)
For ϕ > 0 real non-integer, we obtain
[1− γ1 (k, x)λ
]ϕ−1=
∞∑
j=0
(−1)jΓ(ϕ)Γ(ϕ− j)j!
γ1 (k, x)λj . (15)
Substituting (15) in equation (14) gives
µ′r =λϕαr
Γ(k)
∞∑
j=0
(−1)jΓ(ϕ)Γ(ϕ− j)j!
∫ +∞
0xk+ r
τ−1 exp (−x) γ1 (k, x)λ(1+j)−1 dx. (16)
Using the expansion for γ1 (k, x)λ(1+j)−1 given in Appendix A, we obtain for λ > 0
γ1 (k, x)λ(1+j)−1 =∞∑
l=0
l∑
m=0
(−1)l+m
(λ(1 + j)− 1
l
)(l
m
)γ1 (k, x)m . (17)
13
Substituting (17) in equation (16), µ′r can be expressed as
µ′r =λϕαr
Γ(k)
∞∑
j,l=0
l∑
m=0
wj,l,mI(k,
r
τ,m
), (18)
where
wj,l,m =(−1)j+l+mΓ(ϕ)
Γ(ϕ− j)j!
(λ(1 + j)− 1
l
)(l
m
)
and
I(k,
r
τ,m
)=
∫ +∞
0xk+ r
τ−1 exp (−x) γ1 (k, x)m dx.
For ϕ = 1 and j = 0, we obtain the same result as in Cordeiro et al. (2009). For calculating thelast integral, using the series expansion for the incomplete gamma function yields
I(k,
r
τ,m
)=
∫ ∞
0xk+ r
τ−1 exp(−x)
xk
∞∑
p=0
(−x)p
(k + p)p!
m
dx.
This integral can be determined from equations (24) and (25) of Nadarajah (2008b) in terms of theLauricella function of type A (Exton, 1978; Aarts, 2000) defined by
F(n)A (a; b1, . . . , bn; c1, . . . , cn; x1, . . . , xn) =∞∑
m1=0
· · ·∞∑
mn=0
(a)m1+···+mn (b1)m1· · · (bn)mn
(c1)m1· · · (cn)mn
xm11 · · ·xmn
n
m1! · · ·mn!,
where (a)i is the ascending factorial defined by (with the convention that (a)0 = 1)
(a)i = a(a + 1) · · · (a + i− 1).
Numerical routines for the direct computation of the Lauricella function of type A are available,see Exton (1978) and Mathematica (Trott, 2006). We obtain
I(k,
r
τ,m
)= k−mΓ
(r/τ + k(m + 1)
)×
F(m)A
(r/τ + k(m + 1); k, . . . , k; k + 1, . . . , k + 1;−1, . . . ,−1
). (19)
The moments of the KumGG distribution in (12) and in equations (18) and (19) are the mainresults of this section. Graphical representation of the skewness and kurtosis when α = 0.5, τ = 0.08and k = 3, as a function of λ for selected values of ϕ, and as a function of ϕ for some choices of λ,are given in Figures 4 and 5, respectively.
14
(a) (b)
0 2 4 6 8 10
46
810
1214
16
λ
Ske
wne
ss
ϕ=0.5ϕ=1ϕ=1.5ϕ=2
0 2 4 6 8 10
1020
3040
50λ
Kur
tosi
s
ϕ=0.5ϕ=1ϕ=1.5ϕ=2
Figure 4: Skewness and kurtosis of the KumGG distribution as a function of the parameter λ forselected values of ϕ.
6 Moment Generating Function
Let T be a random variable having the KumGG(α, τ, k, λ, ϕ) density function (3). We nowderive a closed form expression for the mgf, say M(s) = E[exp(sT )], of T . First, we obtain the mgfof the GG(α, τ, k) distribution. We have
Mα,τ,k(s) =τ
ατkΓ(k)
∫ ∞
0exp(st) tτk−1 exp{−(t/α)τ}dt.
Setting u = t/α, we obtain
Mα,τ,k(s) =τ
Γ(k)
∫ ∞
0exp(sαu) uτk−1 exp(−uτ )du.
Expanding the exponential in Taylor series, we have
Mα,τ,k(s) =τ
Γ(k)
∞∑
m=0
(αs)m
m!
∫ ∞
0uτk+m−1 exp(−uτ )du.
15
(a) (b)
0 2 4 6 8 10
24
68
1012
1416
ϕ
Ske
wne
ss
λ=0.5λ=1λ=1.5λ=2
0 2 4 6 8 10
1020
3040
50ϕ
Kur
tosi
s
λ=0.5λ=1λ=1.5λ=2
Figure 5: Skewness and kurtosis of the KumGG distribution as a function of the parameter ϕ forselected values of λ.
The integral is easily obtained as∫∞0 uτk+m−1 exp(−uτ )du = τ−1Γ(k + m/τ) and then
Mα,τ,k(s) =1
Γ(k)
∞∑
m=0
Γ(m
τ+ k
) (αs)m
m!. (20)
Consider the Wright generalized hypergeometric function defined by
pΨq
[ (α1, A1
), · · · ,
(αp, Ap
)(β1, B1
), · · · ,
(βq, Bq)
; x
]=
∞∑
m=0
p∏
j=1
Γ(αj + Aj m)
q∏
j=1
Γ(βj + Bj m)
xm
m!. (21)
By combining equations (20) and (21), we can write
Mα,τ,k(s) =1
Γ(k) 1Ψ0
[(1, 1/τ)− ; sα
], (22)
provided that τ > 1. Clearly, special formulas for the mgf of the distributions listed in Table 1follow immediately from equation (22) by simple substitution of known parameters.
16
The mgf of the KumGG distribution follows by combining equations (11) and (22). We havefor τ > 1
M(s) = 1Ψ0
[(1, 1/τ)− ; sα
] ∞∑
d,j,l,m=0
l∑
q=0
w(d, j, l, q, k, m, λ, ϕ)Γ(k(1 + q) + d)
. (23)
Equation (23) is the main result of this section. The mgf of each KumGG submodel discussed inSection 3 can be determined immediately from equation (23) by substitution of known parameters.
The characteristic function, say φ(t) = E[exp(siT)], corresponding to (23) is simply M(si) forτ > 1, where i =
√−1.
7 Mean Deviations
The amount of scatter in a population is evidently measured to some extent by the totality ofdeviations from the mean and median. If T has the KumGG distribution with density functionf(t), we can derive the mean deviations about the mean µ′1 = E(T ) and about the median m fromthe relations
δ1 =∫ ∞
0| t− µ′1| f(t)dt and δ2 =
∫ ∞
0| t−m| f(t)dt,
respectively. Defining the integral I(s) =∫ s0 tf(t)dt, these measures can be calculated from
δ1 = 2µ′1F (µ′1)− 2I(µ′1) and δ2 = µ′1 − 2I(m), (24)
where F (µ′1) and F (m) are easily obtained from
F (t) = 1−(
1−{
γ1
[k,
(t
α
)τ]}λ)ϕ
.
The integral I(s) can be calculated from equation (11) as
I(s) =∞∑
d,j,l,m=0
l∑
q=0
w(d, j, l, q, k, m, λ, ϕ)J(α, τ, [k(1 + q) + d], s), (25)
whereJ(α, τ, k, s) =
∫ s
0tg(α,τ,k)(t)dt.
We can obtain from the density of the GG(α, τ, k) distribution by setting x = t/α
J(α, τ, k, s) =τα
Γ(k)
∫ s/α
0xτk exp(−xτ )dx.
17
The substitution w = xτ yields J(α, τ, k, s) in terms of the incomplete gamma function
J(α, τ, k, s) =α
Γ(k)
∫ (s/α)τ
0wk+τ−1−1 exp(−w)dw = γ(k + τ−1, (s/α)τ ).
Hence, inserting the last result into (25) yields
I(s) =∞∑
d,j,l,m=0
l∑
q=0
w(d, j, l, q, k, m, λ, ϕ)γ(k(1 + q) + d + τ−1, (s/α)τ ). (26)
The mean deviations of the KumGG distribution follow from (24) and (26).
8 Bonferroni and Lorenz Curves
Bonferroni and Lorenz curves have applications not only in economics to study income andpoverty, but also in other fields like reliability, demography, insurance and medicine. They aredefined by
B(p) =1
pµ′1
∫ q
0tf(t)dx and L(p) =
1µ′1
∫ q
0tf(t)dx,
respectively, where µ′1 = E(T ), q = F−1(p) and I(q) =∫ q0 tf(t)dt. From equation (7) we can obtain
q from a given p. Putting q in equation (26) gives the Bonferroni and Lorenz curves.
9 Order Statistics
The density function fi:n(t) of the ith order statistic, for i = 1, . . . , n, from random variablesT1, . . . , Tn having density (3), is given by
fi:n(t) =1
B(i, n− i + 1)f(t)F (t)i−1{1− F (t)}n−i,
where f(t) and F (t) are the pdf and cdf of the KumGG distribution, respectively and B(·, ·) denotesthe beta function. Using the binomial expansion in the above equation, we readily obtain
fi:n(t) =1
B(i, n− i + 1)f(t)
n−i∑
j1=0
(n− i
j1
)(−1)j1F (t)i+j1−1.
We now present an expression for the density of the KumGG order statistics as a function ofthe baseline density multiplied by infinite weighted sums of powers of G(t). This result enables us
18
to derive the ordinary moments of the KumGG order statistics as infinite weighted sums of theprobability weighted moments (PWMs) of the GG distribution.
From equation (1), we obtain an expansion for F (t)i+j1−1
F (t)i+j1−1 =i+j1−1∑
l1=0
(−1)l1
(i + j1 − 1
l1
) [1−G(t)λ
]l1ϕ.
Using the series expansion for[1−G(t)λ
]l1ϕ
[1−G(t)λ
]l1ϕ=
∞∑
m1=0
(−1)m1
(l1ϕ
m1
)G(t)m1λ,
we have
F (t)i+j1−1 =i+j1−1∑
l1=0
(−1)l1
(i + j1 − 1
l1
) ∞∑
m1=0
(−1)m1
(l1ϕ
m1
)G(t)m1λ.
For m1λ > 0 real non-integer, we can write
G(t)m1λ = {1− [1−G(t)]}m1λ+1−1 =∞∑
l2=0
(−1)l2
(m1λ
l2
)[1−G(t)]m1λ
and then
G(t)m1λ =∞∑
l2=0
l2∑
r=0
(−1)l2+r
(m1λ
l2
)(l2r
)G(t)r.
Replacing∑∞
l2=0
∑l2r=0 by
∑∞r=0
∑∞l2=r gives
G(t)m1λ =∞∑
r=0
∞∑
l2=r
(−1)l2+r
(m1λ
l2
)(l2r
)G(t)r,
where
G(t)m1λ =∞∑
r=0
sr(m1λ)G(t)r
and the coefficients are
sr(m1λ) =∞∑
l2=r
(−1)l2+r
(m1λ
l2
)(l2r
)
19
for r = 0, 1, · · · Further, we have
F (t)i+j1−1 =i+j1−1∑
l1=0
(−1)l1
(i + j1 − 1
l1
) ∞∑
r=0
υr(l1, λ, ϕ)G(t)r, (27)
where the coefficients υr(l1, λ, ϕ) are defined by
υr(l1, λ, ϕ) =∞∑
m1=0
(−1)m1
(l1ϕ
m1
)sr(m1λ).
Interchanging the sums, we obtain
F (t)i+j1−1 =∞∑
r=0
pr,i+j1−1(λ, ϕ)G(t)r,
where the coefficients pr,u(λ, ϕ) can be calculated as
pr,u(λ, ϕ) =u∑
l1=0
(−1)l1
(u
l1
) ∞∑
m1=0
∞∑
l2=r
(−1)m1+r+l2
(l1ϕ
m1
)(m1λ
l2
)(l2r
)
for r, u = 0, 1, · · · The density of the KumGG order statistics is given by
fi:n(t) =1
B(i, n− i + 1)f(t)
n−i∑
j1=0
(−1)j1
(n− i
j1
) ∞∑
r=0
pr,i+j1−1(λ, ϕ)G(t)r. (28)
Alternatively, we write
fi:n(t) =1
B(i, n− i + 1)
∞∑
d,j,l,m,r=0
l∑
q=0
n−i∑
j1=0
(−1)j1
(n− i
j1
)w(d, j, l, q, k, m, λ, ϕ)pr,i+j1−1(λ, ϕ)
× γ1
[k,
(t
α
)τ]r
g(α,τ,k(1+q)+d)(t). (29)
A general theory for PWMs covers the summarization and description of theoretical probabilitydistributions. The PWM method can generally be used for estimating parameters of a distributionwhose inverse form cannot be expressed explicitly.
The (s, r)th PWM of T following the KumGG distribution , say δs,r, is formally defined by
δs,r = E{T sF (T )r} =∫ ∞
0tsF (t)rf(t)dt.
20
An alternative formula for moments of order statistics is given by
E (T si:n) =
∫ ∞
0tsfi:n(t)dt
Equation (29) can be rewritten as
fi:n(t) =∞∑
d,j,l,m,r=0
l∑
q=0
n−i∑
j1=0
w1(d, i, j, j1, l, q, k,m, n, τ, λ, ϕ)tτ(kq+d)γ1
[k,
(t
α
)τ]r
g(α,τ,k)(t),
where w1(d, i, j, j1, l, q, k,m, n, τ, λ, ϕ) = (−1)j+j1+l+qλϕΓ(ϕ)sm(λ)cq,d
B(i,n−i+1)ατ(kq+d)Γ(k)qΓ(ϕ−j)j!
(n−ij1
)(λj+m
l
)(lq
). Therefore,
the moments of the KumGG order statistics can be expressed by
E (T si:n) =
∞∑
d,j,l,m,r=0
l∑
q=0
n−i∑
j1=0
w1(d, i, j, j1, l, q, k, m, n, τ, λ, ϕ)pr,i+j1−1(λ, ϕ)δs+τ(kq+d),r.
10 Maximum Likelihood Estimation
Let Ti be a random variable following (3) with the vector of parameters θ = (α, τ, k, λ, ϕ)T .The data encountered in survival analysis and reliability studies are often censored. A very simplerandom censoring mechanism that is often realistic is one in which each individual i is assumedto have a lifetime Ti and a censoring time Ci, where Ti and Ci are independent random variables.Suppose that the data consist of n independent observations ti = min(Ti, Ci) for i = 1, · · · , n. Thedistribution of Ci does not depend on any of the unknown parameters of Ti. Parametric inferencefor such data are usually based on likelihood methods and their asymptotic theory. The censoredlog-likelihood l(θ) for the model parameters is
l(θ) = r log
[λϕτ
αΓ(k)
]−
∑
i∈F
(tiα
)τ
+ (τk − 1)∑
i∈F
log(
tiα
)+ (λ− 1)
∑
i∈F
log
{γ1
[k,
(tiα
)τ]}
+(ϕ− 1)∑
i∈F
log
(1−
{γ1
[k,
(tiα
)τ]}λ)
+ϕ∑
i∈C
log
(1−
{γ1
[k,
(tiα
)τ]}λ)
, (30)
where r is the number of failures and F and C denote the uncensored and censored sets of obser-vations, respectively.
The score components corresponding to the parameters in θ are given by
Uα(θ) = −rτk
α+
τ
α
∑
i∈F
ui − τ
α
∑
i∈F
visi +λτ(ϕ− 1)
αΓ(k)
∑
i∈F
uipi +λτϕ
αΓ(k)
∑
i∈C
uipi,
21
Uτ (θ) =r
τ− 1
τ
∑
i∈F
ui log(ui) +k
τ
∑
i∈F
log(ui) +1τ
∑
i∈F
visi log(ui)− λ(ϕ− 1)τ
∑
i∈F
vipi log(ui)
−λϕ
τ
∑
i∈C
vipi log(ui),
Uk(θ) = −rλψ(k) +∑
i∈F
log(ui) +∑
i∈F
siqi + (ϕ− 1)[rλψ(k)]∑
i∈F
piγ1(k, ui)− λ(ϕ− 1)∑
i∈F
piqi
+λϕψ(k)(n− r − 1)∑
i∈C
piγ1(k, ui)− λϕ∑
i∈F
piqi,
Uλ(θ) =r
λ+
∑
i∈F
log[γ1(k, ui)]− (ϕ− 1)∑
i∈F
bi[γ1(k, ui)]λ − ϕ∑
i∈C
bi[γ1(k, ui)]λ
and
Uϕ(θ) =r
ϕ+
n∑
i=1
log(ωi),
where
ui =(
tiα
)τ
, gi = uki exp(−ui), ωi = 1− γ1(k, ui)λ, vi =
gi
Γ(k), si =
(λ− 1)γ1(k, ui)
,
[γ(k, ui)]k =∞∑
n=0
(−1)n
n!J(ui, k + n− 1, 1),
pi =γ1(k, ui)λ−1
ωi, qi =
[γ(k, ui)]kΓ(k)
, bi =log[γ1(k, ui)]
ωi,
ψ(.) is the digamma function and J(ui, k + n− 1, 1) is defined in Appendix C.For interval estimation and hypothesis tests on the model parameters, we require the 5× 5 unit
observed information matrix
J = J(θ) =
jα,α jα,τ jα,k jτ,λ jα,ϕ
jτ,α jτ,τ jτ,k jτ,λ jτ,ϕ
jk,α jk,τ jk,k jk,λ jk,ϕ
jλ,α jλ,τ jλ,k jλ,λ jλ,ϕ
jϕ,α jϕ,τ jϕ,k jϕ,λ jϕ,ϕ
,
22
whose elements are given in Appendix C. Consequently, the MLE θ of θ is obtained numericallyfrom the nonlinear equations
Uα(θ) = Uτ (θ) = Uk(θ) = Uλ(θ) = Uϕ(θ) = 0.
Under conditions that are fulfilled for parameters in the interior of the parameter space but noton the boundary, the asymptotic distribution of
√n(θ − θ) is N5(0, I(θ)−1),
where I(θ) is the unit expected information matrix. This approximated distribution holds whenI(θ) is replaced by J(θ), i.e., the observed information matrix evaluated at θ. The multivariatenormal N5(0, J(θ)−1) distribution can be used to construct approximate confidence intervals for theindividual parameters and for the hazard and survival functions. In fact, an asymptotic confidenceinterval with significance level γ for each parameter θr is given by
ACI(θr, 100(1− γ)%) =(
θr − zγ/2
√jθr,θr
n, θr + zγ/2
√jθr,θr
n
),
where jθr,θr is the estimated rth diagonal element of J(θ)−1 for r = 1, . . . , 5 and zγ/2 is the quantile1− γ/2 of the standard normal distribution.
The likelihood ratio (LR) statistic is useful for testing goodness-of-fit of the KumGG distributionand for comparing it with some of its special sub-models (see Section 3). We can compute themaximum values of the unrestricted and restricted log-likelihoods to construct LR statistics fortesting some sub-models of the KumGG distribution. For example, we may use the LR statistic tocheck if the fit using the KumGG distribution is statistically “superior” to a fit using the KumGHN,KumSChi, GG and KumW distributions for a given data set. In any case, hypothesis tests of thetype H0 : θ = θ0 versus H : θ 6= θ0, where θ0 is a specified vector, can be performed using anyof the above three asymptotically equivalent statistics. For example, the test of H0 : ϕ = 1 versusH : H0 not true is equivalent to compare the EGG distribution with the KumGG distribution forwhich the LR statistic reduces to
w = 2[`(α, τ , k, λ, ϕ)− `(α, τ , k, λ, 1)
],
where α, τ , k, λ and ϕ are the MLEs under H and α, τ , k and λ are the estimates under H0.
11 Applications
In this section, we illustrate the usefulness of the KumGG distribution.
23
11.1 Simulation study
We conducted Monte Carlo simulation studies to assess on the finite sample behavior of themaximum likelihood estimators of α, τ , k, λ and ϕ. All results were obtained from 3000 MonteCarlo replications and the simulations were carried out using the statistical software package R. Ineach replication a random sample of size n is drawn from the KumGG(α, τ, k, λ, ϕ) distribution andthe BFGS method (see, for example, Press et al., 1992) has been used by the authors for maximizingthe total log-likelihood function l(θ). KumGG random number generation was performed usingthe inversion method. The true parameter values used in the data generating processes are α = 0.2,τ = 0.2, k = 2, λ = 1.3 and ϕ = 2. Table 2 presents the mean maximum likelihood estimates of thefour parameters that index the KumGG distribution along with the respective root mean squarederrors (RMSE) for sample sizes n = 50, n = 80 and n = 100.
Table 2: Mean estimates and root mean squared errors of α, τ , k and λ, the maximum likelihoodestimators of the EGG parameters
n Paremeter Mean RMSE50 α 0.4051 1.1824
τ 0.4202 1.0227k 2.3286 1.6605λ 1.1481 1.2023ϕ 1.9605 0.8362
80 α 0.1992 0.7449τ 0.2352 0.3338k 0.2164 1.4469λ 1.0610 0.8409ϕ 2.0041 0.8669
100 α 0.1650 0.6964τ 0.2212 0.2921k 2.0636 1.2536λ 1.1037 0.8396ϕ 2.0441 0.8066
The figures in Table 2 we notice that the biases and root mean squared errors of the maximumlikelihood estimators ofα, τ , k, λ and ϕ decay toward zero as the sample size increases, as expected.We also note that there is small sample bias in the estimation of the parameters that index theKumGG distribution. Future research should obtain bias corrections for these estimators.
24
11.2 Aarset data - uncensored
Here, we illustrate the superiority of the new distribution as compared to some of its submodelsand also to the alternative EGG, GG, Weibull and beta Weibull (BW) distributions. The BWmodel is not a submodel of the KumGG distribution. The BW distribution was introduced byFamoye et al. (2005) and its cdf is
F (t) =1
B(a, b)
∫ {1−exp[−(t/α)γ ]}
0wa−1(1− w)b−1dw.
We consider the widely used data from Aarset (1987), and also reported in Mudholkar et al. (1996)and Wang (2000), on lifetimes of 50 components. Table 3 gives the MLEs (and the correspondingstandard errors in parentheses) of the model parameters and the values of the following statisticsfor some models: AIC (Akaike Information Criterion), BIC (Bayesian Information Criterion) andCAIC (Consistent Akaike Information Criterion). The AIC and BIC values for the KumGG modelare the smallest values among those of the five fitted models, and hence our new model can bechosen as the best model.
Table 3: Estimates of the model parameters for Aarset data, the corresponding SEs (given inparentheses) and the statistics AIC, BIC and CAIC.
Model α τ k λ ϕ AIC BIC CAICKumGG 84.5056 79.5358 0.0080 0.5393 0.3431 423.1 432.7 424.5
(0.2099) (2.0929) (0.0021) (0.2387) (0.0565)EGG 86.7690 28.8487 1.0314 0.0272 1 455.7 463.4 456.6
(0.5539) (0.0308) (0.0001) (0.0039) (−)GG 86.9241 259.01 0.0028 1 1 446.7 452.4 447.2
(0.8946) (0.0024) (0.0004) (−) (−)Weibull 44.9126 0.9490 1 1 1 486.0 489.8 486.3
(6.6451) (0.1196) (−) (−) (−)α γ a b
Beta Weibull 49.6326 5.9441 0.0783 0.0702 (−) 444.5 452.1 445.4(3.7606) (0.1394) (0.0166) (0.0288) (−)
In order to assess if the model is appropriate, Figure 6 plots the empirical survival functionand the estimated survival function of the KumGG distribution. It is a very competitive model fordescribing the bathtub-shaped failure rate of the Aarset data.
The plots of the estimated densities and the histogram of the Aarset data given in Figure 7show that the KumGG distribution produces a better fit than the other four models.
25
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
t
S(t)
Kaplan−MeierKumGGEGGGGWeibullBeta Weibull
Figure 6: Estimated survival function by fitting the KumGG distribution and some other modelsand the empirical survival for the Aarset data.
11.3 Serum reversal data - censored
The data set refers to the serum-reversal time (days) of 148 children contaminated with HIVfrom vertical transmission at the University Hospital of the Ribeirao Preto School of Medicine(Hospital das Clınicas da Faculdade de Medicina de Ribeirao Preto) from 1986 to 2001 (Silva, 2004;Perdona, 2006; Carrasco, et al., 2008). We assume that the lifetimes are independently distributed,and also independent from the censoring mechanism. We assume right-censored lifetime data(censoring random). Table 4 gives the MLEs (and the corresponding standard errors in parentheses)of the parameters and the values of the AIC, BIC and CAIC statistics. These results indicate thatthe KumGG model has the lowest AIC, BIC and CAIC values among those of all fitted models,and hence it could be chosen as the best model.
In order to assess if the model is appropriate, plots of the empirical and estimated survivalfunctions of the KumGG, EGG, GG, Weibull and BW distributions are given in Figure 8. We
26
t
f(t)
0 20 40 60 80
0.00
0.01
0.02
0.03
0.04
0.05
KumGGEGGGGWeibullBeta Weibull
Figure 7: Estimated densities of the KumGG, EGG, GG, Weibull and beta Weibull models for theAarset data.
conclude that the KumGG distribution provides a good fit for these data.
12 Conclusions
We study a four parameter lifetime distribution, so-called “the Kum-generalized gamma (KumGG)distribution” which is a simple extension of the generalized gamma distribution (Stacy, 1962).The new model generalizes several distributions widely used in the lifetime literature and is moreflexible than the exponentiated generalized gamma, generalized gamma, generalized half-normal,exponentiated Weibull, among others distributions. The proposed distribution could have increas-ing, decreasing, bathtub and unimodal hazard rate functions. It is then very versatile to modellifetime data with a bathtub shaped hazard rate function and also to model a variety of uncer-tainty situations. We provide a mathematical treatment of this distribution including the order
27
Table 4: Estimates of the model parameters for the serum reversal data, the corresponding SEs(given in parentheses) and the statistics AIC, BIC and CAIC.
Model α τ k λ ϕ AIC BIC CAICKumGG 350.05 49.8303 0.2176 0.1282 0.3424 770.7 785.7 771.1
(1.5707) (5.8895) (0.0073) (0.0236) (0.0522)EGG 350.45 22.2991 1.0741 0.1072 1 798.1 810.1 798.3
(2.4187) (0.0375) (0.0004) (0.0113) (−)GG 379.40 24.5312 0.0974 1 1 783.7 792.7 783.9
(8.8211) (10.3258) (0.0402) (−) (−)Weibull 307.62 3.1132 1 1 1 808.0 814.0 808.1
(12.3523) (0.3250) (−) (−) (−)α γ a b
Beta Weibull 349.99 6.3895 0.3944 0.9273 (−) 797.9 809.9 798.2(23.0923) (0.7657) (0.0468) (0.3361) (−)
statistics. Explicit closed form expressions for the moments and moment generating function areprovided which hold in generality for any parameter values, we obtain infinite weighted sums forthe moments of the order statistics. The application of the proposed distribution is straightforward.We discuss maximum likelihood estimation and testing of hypotheses for the model parameters.The KumGG distribution allows goodness-of-fit tests of some well-known distributions in reliabilityanalysis by taking these distributions as sub-models. The practical relevance and applicability ofthe new model are demonstrated in two real data sets. We verify that some special sub-modelscould provide similar fits while being more parsimonious models. The applications demonstrate theusefulness of the KumGG distribution, and with the use of modern computer resources with ana-lytic and numerical capabilities, it can be an adequate tool comprising the arsenal of distributionsfor lifetime data analysis.
28
0 50 100 150 200 250 300 350
0.0
0.2
0.4
0.6
0.8
1.0
t
S(t)
Kaplan−MeierKumGGEGGGGWeibullBeta Weibull
Figure 8: Estimated survival function by fitting the KumGG distribution and some other modelsand the empirical survival for the serum reversal data.
Appendix A
As discuss before, we derive an expansion for γ1
[k,
(tα
)τ ]λ−1 for any λ > 0 real non-integer.We can write
γ1
[k,
(t
α
)τ]λ−1
=(
1−{
1− γ1
[k,
(t
α
)τ]})λ−1
=∞∑
j=0
(λ− 1
j
)(−1)j
{1− γ1
[k,
(t
α
)τ]}j
,
which always converges since 0 < γ1
[k,
(tα
)τ ]< 1. Hence,
γ1
[k,
(t
α
)τ]λ−1
=∞∑
j=0
j∑
m=0
(−1)j+m
(λ− 1
j
)(j
m
)γ1
[k,
(t
α
)τ]m
. (31)
29
We can substitute∑∞
j=0
∑jm=0 for
∑∞m=0
∑∞j=m to obtain
γ1
[k,
(t
α
)τ]λ−1
=∞∑
m=0
∞∑
j=m
(−1)j+m
(λ− 1
j
)(j
m
)γ1
[k,
(t
α
)τ]m
.
and then
γ1
[k,
(t
α
)τ]λ−1
=∞∑
m=0
sm(λ)γ1
[k,
(t
α
)τ]m
, (32)
where
sm(λ) =∞∑
j=m
(−1)j+m
(λ− 1
j
)(j
m
). (33)
We now use the series expansion for the incomplete gamma ratio function given by
γ1
[k,
(t
α
)τ]=
1Γ(k)
(t
α
)τk ∞∑
d=0
[−
(t
α
)τ]d 1(k + d)d!
. (34)
By application of an equation in Section 0.314 of Gradshteyn and Ryzhik (2000) for power seriesraised to powers, we obtain for any q positive integer
[ ∞∑
d=0
ad
(t
α
)τd]q
=∞∑
d=0
cq,d
(t
α
)τd
, (35)
where the coefficients cq,d (for d = 1, 2, · · · ) satisfy the recurrence relation
cq,d = (da0)−1d∑
p=1
(qp− d + p)apcq,d−p, (36)
cq,0 = aq0 and ap = (−1)p/(k + p)p!. The coefficient cq,d can be calculated from cq,0, . . . ,
cq,d−1, and it can be written explicitly as functions of the quantities a0, . . . , ad, although it is notnecessary for programming numerically our expansions. Further, using equation (35), we obtain
γ1
[k,
(t
α
)τ]q
=1
Γ(k)q
(t
α
)τkq ∞∑
d=0
cq,d
(t
α
)τd
(37)
where the coefficients cq,d are just obtained from equation (36).
Appendix B
30
Table 5: Some Special Sub-Models of the KumGG distribution.
Distribution α τ k λ ϕ Author
Exponentiated - - - - 1 Cordeiro et al. (2009)generalized gamma
Exponentiated - 1 1 - 1 Gupta and Kundu (1999)exponential Gupta and Kundu (2001)
Kum-Rayleigh - 2 1 - - New
Generalized Rayleigh - 2 1 - 1 Kundu and Raqab (2005)
Kum-Weibull - - 1 - - Cordeiro and Castro (2010)
Exponentiated - - 1 - 1 Mudholkar et al. (1995)Weibull Mudholkar et al. (1996)
Weibull - - 1 1 1 Mudholkar et al. (1995)
Kum-Exponential - 1 1 - - Cordeiro and Castro (2010)
Exponential - 1 1 1 1 Steffensen (1930)
Rayleigh - 2 1 1 1 Siddiqui (1962)
Kum-Gamma - 1 - - - Cordeiro and Castro (2010)
Exponentiated-gamma with - 1 - - 1 Gupta et al. (1998)three parameters
Exponentiated-Gamma with - 1 - - 1 Gupta et al. (1998)two parameters
Kum-Gamma with - 1 1 - - Newone parameter
Gamma with two parameters - 1 - 1 1 Brow and Flood (1947)
Gamma with one parameter - 1 1 1 1 Jambunathan (1954)
Kum-Chi-Square 2 2 p/2 - - New
Exponentiated Chi-Square 2 2 p/2 - 1 New
Chi-Square 2 2 p/2 1 1 Fisher (1924)
Kum-Scaled Chi-Square√
2σ 1 p/2 - - New
Exponentiated Scaled√
2σ 1 p/2 - 1 NewChi-Square
Kum-Maxwell√
θ 2 3/2 - - New
Exponentiated Maxwell√
θ 2 3/2 - 1 New
Maxwell√
θ 2 3/2 1 1 Bekker and Roux (2005)
Kum-Nakagami√
w/µ 2 µ - - New
Exponentiated Nakagami√
w/µ 2 µ - 1 New
Nakagami√
w/µ 2 µ 1 1 Shankar et al. (2001)
Kum-Generalized 212γ 2γ 1/2 - - New
Half-Normal
Exponentiated Generalized 212γ 2γ 1/2 - 1 New
Half-Normal
Kum-Half-Normal 212 θ 2 2 - - New
Exponentiated Half-Normal 212 θ 2 1/2 - 1 New
Generalized Half-Normal 212γ θ 2γ 1/2 1 1 Cooray and Ananda (2008)
Half-Normal 212 θ 2 1/2 1 1 Hogg and Tanis (1993)
31
Appendix C
Upon differentiating (30), the elements of the observed information matrix J(θ) for the para-meters (α, τ, k, λ, ϕ) are
Jαα(θ) =rτk
α2− τ(1− τ)
α2
∑
i∈F
ui − τ
α2
∑
i∈F
visi
(−1 +
τ
γ1(k, ui){γ1(k, ui)[−k + ui] + vi}
)
+λτ(ϕ− 1)
α2
∑
i∈F
vipi
(−1 +
τ
ωi{ωi(−k + ui − visi)− λvi[γ1(k, ui)]2}
)
+λτϕ
α2
∑
i∈F
vipi
(−1 +
τ
ωi{ωi(−k + ui − visi)− λvi[γ1(k, ui)]2}
),
Jατ (θ) = −rk
α+
1α
∑
i∈F
ui [1 + log(ui)]− 1α
∑
i∈F
visi
(1 +
log(ui)γ1(k, ui)
{γ1(k, ui)[k − ui]− vi})
+λ(ϕ− 1)
α
∑
i∈F
vipi
(1 +
log(ui)ωi
{ωi(k − ui + vipi) + λvi[γ1(k, ui)]2
})
+λ
α
∑
i∈C
vipi
(1 +
log(ui)ωi
{ωi(k − ui + vipi) + λvi[γ1(k, ui)]2
}),
Jαk(θ) = −rτ
α− τ
α
∑
i∈F
visi
[log(ui)− qi
γ1(k, ui)
]+
λτ(ϕ− 1)α
∑
i∈F
vipi
(−ψ(k) +
1ωi
(ωi{log(ui) + si[−ψ(k)γ1(k, ui) + qi]}) + {λωipi[−ψ(k)γ1(k, ui) + qi]})
+λτϕ
α
∑
i∈C
vipi
(−ψ(k) +
1ωi
(ωi{log(ui) + si[−ψ(k)γ1(k, ui) + qi]})
+ {λωipi[−ψ(k)γ1(k, ui) + qi]})
,
Jαλ(θ) = − τ
α
∑
i∈F
vi
γ1(k, ui)+
τ(ϕ− 1)α
∑
i∈F
vipi
(1 + λbi
{ωi + [γ1(k, ui)]λ
})
+τϕ
α
∑
i∈C
vipi
(1 + λbi
{ωi + [γ1(k, ui)]λ
}),
32
Jαϕ(θ) =λτ
α
n∑
i=1
vipi, Jτϕ(θ) = −λ
τ
n∑
i=1
vipi log(ui),
Jττ (θ) = − r
τ2− 1
τ2
∑
i∈F
ui[log(ui)]2 +1τ2
∑
i∈F
visi[log(ui)]2
γ1(k, ui)[γ1(k, ui)(k − ui)− vi]
−λ(ϕ− 1)τ2
∑
i∈F
vipi[log(ui)]2
ωi{ωi[(k − ui) + visi] + λvi[γ1(k, ui)]2}
−λϕ
τ2
∑
i∈C
vipi[log(ui)]2
ωi{ωi[(k − ui) + visi] + λvi[γ1(k, ui)]2},
Jτk(θ) =1τ
∑
i∈F
log(ui) +1τ
∑
i∈F
visi log(ui)[log(ui)− qi
γ1(k, ui)
]− λ(ϕ− 1)
τ
∑
i∈F
vipi log(ui){−ψ(k) + log(ui) +
[−ψ(k) +
qi
γ1(k, ui)
][λ− 1 + λpiγ1(k, ui)]
}
−λϕ
τ
∑
i∈C
vipi log(ui){−ψ(k) + log(ui) +
[−ψ(k) +
qi
γ1(k, ui)
][λ− 1 + λpiγ1(k, ui)]
},
Jτλ(θ) =1τ
∑
i∈F
vi log(ui)γ1(k, ui)
− (ϕ− 1)τ
∑
i∈F
vipi log(ui){1 + λbiωi[1 + piγ1(k, ui)]}
−ϕ
τ
∑
i∈C
vipi log(ui){1 + λbiωi[1 + piγ1(k, ui)]},
Jkk(θ) = −rλψ′(k) +1
Γ(k)
∑
i∈F
si
{[γ(k, ui)]kk −
([γ(k, ui)]k)2
γ1(k, ui)
}
+rλ(ϕ− 1)∑
i∈F
piγ1(k, ui){
ψ′(k) +λψ(k)
γ1(k, ui)
[−ψ(k)γ1(k, ui)
ωi+
qi
ωi
]}− λ(ϕ− 1)
Γ(k)
∑
i∈F
pi
(−ψ(k) [γ(k, ui)]k +
{qi(λ− 1)
[−ψ(k)γ1(k, ui) +
qi
γ1(k, ui)
]+
[γ(k, ui)]kk
ωi
})
+λϕ(n− r − 1)∑
i∈C
piγ1(k, ui){
ψ′(k) +λψ(k)
γ1(k, ui)
[−ψ(k)γ1(k, ui)
ωi+
qi
ωi
]}− λϕ
Γ(k)
∑
i∈C
pi
(−ψ(k) [γ(k, ui)]k +
{qi(λ− 1)
[−ψ(k)γ1(k, ui) +
qi
γ1(k, ui)
]+
[γ(k, ui)]kk
ωi
}),
33
Jkλ(θ) = −rψ(k) +∑
i∈F
qi
γ1(k, ui)+ rψ(k)(ϕ− 1)
∑
i∈F
piγ1(k, ui)(1 + λbi)− (ϕ− 1)
∑
i∈F
piqi(1 + λbi) + ϕψ(k)(n− r − 1)∑
i∈C
piγ1(k, ui)(1 + λbi)− ϕ∑
i∈C
piqi(1 + λbi),
Jkϕ(θ) = rλψ(k)∑
i∈F
piγ1(k, ui) + λψ(k)(n− r − 1)∑
i∈C
piγ1(k, ui)− λ
n∑
i=1
piqi,
Jλλ(θ) = − r
λ2− (ϕ− 1)
∑
i∈F
bi[γ1(k, ui)]λ log[γ1(k, ui)]− ϕ∑
i∈C
bi[γ1(k, ui)]λ log[γ1(k, ui)],
Jλϕ(θ) = −n∑
i=1
bi[γ1(k, ui)]λ, Jϕϕ(θ) = − r
ϕ2,
where
[γ(k, ui)]k =∞∑
n=0
(−1)n
n!J(ui, k + n− 1, 1), [γ(k, ui)]kk =
∞∑
n=0
(−1)n
n!J(ui, k + n− 1, 2),
ψ′(.) is the derivative of the digamma function, ui, gi, ωi, vi, si, pi, qi and bi are defined in Section10. The J(., ., .) function can be easily calculated from the integral given by Prudnikov et al. (1986,vol 1, Section 2.6.3, integral 1)
J(a, p, 1) =∫ a
0xp log(x)dx =
ap+1
(p + 1)2[(p + 1) log(a)− 1].
and
J(a, p, 2) =∫ a
0xp log2(x)dx =
ap+1
(p + 1)3{2− (p + 1) log(a)[2− (p + 1) log(a)]}.
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