The Gamma-Weibull-G Family of Distributions with Applications

Weibull distribution and its extended families has been widely studied in lifetime applications. Based on the Weibull-G family of distributions and the exponentiated Weibull distribution, we study in detail this new class of distributions, namely, Gamma-WeibullG (GWG) family of distributions. Some special models in the new class are discussed. Statistical properties of the family of distributions, such as expansion of density function, hazard and reverse hazard functions, quantile function, moments, incomplete moments, generating functions, mean deviations, Bonferroni and Lorenz curves and order statistics are presented. We also present Rényi entropy, estimation of parameters by using method of maximum likelihood, asymptotic confidence intervals and applications using real data sets.


Introduction
Numerous classical distributions have been extensively discussed over the past years for modeling data in various areas such as economics, finance, insurance, actuarial, engineering, demography, biological studies, environmental and medical sciences.However, in many of those applied areas, there is more and more demands for extended forms of these distributions.Hence, several methods for generating new families of distributions have been studied in recent years.Researchers often define new families of probability distributions that extend well-known distributions in order to provide great flexibility in modeling data in practice.Recently, (Alzaatreh and Ghosh 2014) introduced and studied the Weibull-X family of distributions, (Alzaatreh, Famoye, and Lee 2014) developed the properties of the gamma-X family of distributions in general and studied the gamma-normal distribution as a special case, and (Nascimento, Bourguignon, Zea, Santos-Neto, Silva, and Cordeiro 2014) introduced the gamma extended Weibull family of distributions.Moreover, (Nadarajah, Cordeiro, and Ortega 2015) provided a comprehensive analysis of general mathematical and statistical properties of Zografos and Balakrishnan-G distributions.See references therein.Most recently, (Tahir, Zubair, Mansoor, Cordeiro, Alizadeh, and Hamedani 2015) introduced a new generator based on the Weibull random variable, namely, the new Weibull-G family.
We introduce a new extended generator, the gamma-Weibull-G family based on their pre-vious work and combined the gamma-generator with the Weibull-G family of distributions which was defined by (Bourguignon, Silva, and Cordeiro 2014).We hope the new family of distributions yields a better fit in certain practical situations.Additionally, we provide a rigorous and comprehensive account of the mathematical properties of the proposed family of distributions.
The results in this paper are organized in the following manner.The new model called gamma Weibull-G (GWG) distribution and its sub-models are given in section 2. In section 3, statistical properties including expansion of the probability density function, hazard rate, reverse hazard rate and quantile functions, moments, conditional moments, moment generating and characteristics functions, mean deviations, Lorenz and Bonferroni curves, order statistics and Rényi entropy are presented.Maximum likelihood estimates of the model parameters and observed information matrix are given in section 4. The special cases of gamma-Weibull-Uniform (GWU) and gamma-Weibull-Weibull (GWW) distributions are presented in details in sections 5 and 6, respectively.A Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimates are presented in section 7. Section 8 contains applications of the new model to real data sets.A short concluding remark is given in section 9.

The model
Considering a continuous cumulative distribution function (cdf) G(x) with probability density function (pdf) g(x) and survival function Ḡ(x) = 1 − G(x), (Bourguignon et al. 2014) defined the Weibull-G family of distribution with cdf: where θ is a vector of parameters.The pdf of this family of distributions is given by Therefore, given a continuous baseline distribution G(x), one can derive the Weibull-G distribution with two extra parameters α and β from pdf (2).In the rest of this section, we consider the combination of Weibull-G (W-G) distribution with distribution proposed by (Zografos and Balakrishnan 2009) with cdf, where γ(δ, x) = x 0 t δ−1 e −t dt is the lower incomplete gamma function and Γ (•) is the gamma function.The corresponding pdf is given by By taking G(x) = F W G (x) in (3), we come up with the gamma-Weibull-G family of distributions (GWG), with cdf: and pdf Therefore, for each G distribution, we define the Gamma-Weibull-G (GWG) distribution with three extra parameters α, β and δ by the pdf (6).A random variable X with pdf (6) is denoted by X ∼ GW G (α, β, δ, θ).The additional parameters induced by the Gamma-Weibull (G-W) generator induces a family of more flexible distributions.An interpretation of the GWG family of distributions can be given as follows (Bourguignon et al. 2014) in a similar context.Let Z be a lifetime random variable having a certain continuous G distribution.The odds ratio that an individual (or component) following the lifetime Z will die (failure) at time x is G(x; θ)/ Ḡ(x; θ).Consider that the variability of this odds of death is represented by the random variable X and assume that it follows the Gamma-Weibull model with scale parameter α, and shape parameters δ and β.We can write P r(Z < x) = P r X ≤ α G(x; θ) Ḡ(x; θ) which is given by (5).To avoid issues or problems with over parametrization and redundancy, we restrict the parameter vector θ to at most a two component vector.

Sub-models of the GWG distribution
If δ = 1, the GWG distribution corresponds to the Weibull-generator, which was proposed by (Bourguignon et al. 2014).When β = 1, and δ = β = 1, GWG distribution reduces to the gamma-exponential-generator, and exponential-generator (Cordeiro, Ortega, and Cunha 2013b), respectively.Moreover, β = 2, and β = 2, δ = 1 lead to the gamma-Rayleighgenerator, and Rayleigh-generator, respectively.In addition, a lot of sub-models can be obtained when we change the parameter vector θ and these special cases as well as the corresponding parameters are listed in Table 1.

Statistical properties of the GWG distribution
Some statistical properties, such as expansion of density function, hazard and reverse hazard functions, quantile function, moments, incomplete moments, generating functions, mean deviations, Bonferroni and Lorenz curves and distribution of order statistics of the GWG distribution are presented in this section.
The Gamma-Weibull-G Family of Distributions with Applications

Expansion of the pdf of GWG distribution
Considering the power series for exponential function, we can obtain the following equation: Inserting this equation into (6), we obtain We obtain where and denote the pdf of the exponentiated generalized (EG) (Cordeiro et al. 2013b) distribution with parameter β * = β(δ + i) + j.The expansion of the pdf of GWG distribution illustrates that the GWG density function is indeed an infinite linear combination of EG density functions.Thus, a lot of the mathematical and statistical properties of the GWG distribution come directly from those of EG distribution.

Hazard, reverse hazard and quantile functions
In this section, we provide the hazard, reverse hazard and quantile functions of the GWG distribution.

Hazard and reverse hazard functions
Note that if X is a continuous random variable with cdf G(x), and pdf g(x), then the hazard rate function (hrf), reverse hazard function (rhf) and mean residual life functions are given by h respectively.The functions h G (x), δ G (x), and G(x) are equivalent.See (Shaked and Shanthikumar 1994) for additional details.The hazard rate and reverse hazard rate functions of GWG distribution are given by and respectively.

Quantile function
The GWG quantile function can be obtained by inverting F GW G (x) = u, where F GW G (x) is given by ( 5).Hence, we obtain the following equation: Thus, the quantile x u of the GWG(α, β, δ, θ) reduces to the real solution of the following equation: that is, the quantile x u of the GWG(α, β, δ, θ) can be derived from the quantile x q of the baseline distribution with cdf G(x; θ) and it is given by where q is defined in equation ( 14).

Moments, incomplete moments, moment generating and characteristic functions
Let X ∼ GW G(α, β, δ, θ), the s th moment of X can be obtained from (9) as where E(Z s i,j ) denotes the s th moment of Z i,j , which follows the EG distribution with the parameter β * = β(δ + i) + j and ω i,j is given by equation (10).Similarly, the incomplete moments and the moment generating function (mgf) can be obtained as follows: ω i,j E(e tZ i,j ), where E(e tZ i,j ) is the mgf of the EG distribution with parameter β * = β(δ + i) + j and ω i,j is given by (10).The characteristic function is given by φ(t) = E(e itX ), where i = √ −1, thus one can obtain with φ β(δ+i)+j (t) denotes the characteristic function of EG distribution, where parameter β * = β(δ + i) + j, and ω i,j is defined as (10).
Since it is always of interest to know conditional expectations for lifetime models, we present it as follow: where I i,j (y) = y 0 x t h β(δ+i)+j (x, θ)dx, and ω i,j is given by (10).

Mean deviation, Bonferroni and Lorenz curves
Let X ∼ GW G(α, β, δ, θ), the mean deviation about the mean and the mean deviation about the median are defined by , respectively, where µ = E(X) and M = M edian(X) denotes the median.They can be expressed as where m(z) = z 0 xf GW G (x)dx is the first incomplete moment.These quantities have been applied to a wide variety of fields, such as studying of income and property in economics, reliability, demography, insurance and medicine.Bonferroni and Lorenz curves are defined by respectively, where µ = E(X) and q = F −1 GW G (p) is obtained from equation ( 14) and ( 15).Using similar methods in deriving the moments, we can show that ω i,j I β(δ+i)+j (i, j), and ω i,j I β(δ+i)+j (i, j), where I β(δ+i)+j (i, j) = q 0 xf β(δ+i)+j (x, θ)dx, is the first incomplete moment of the EG distribution with parameter β * = β(δ + i) + j and ω i,j is given by (10).

Order statistics
In this section, distribution of the i th order statistic for the GWG distribution is presented.Let X 1 , . . ., X n be independent and identically distributed (i.i.d) random variables distributed according to (6).The pdf of the i th order statistic, for example X i:n , is given by Using the binomial theorem, the pdf of i th order statistic can be written as Note that for the incomplete gamma function γ(x, δ), we apply the following power series (see (Gradshteyn and Ryzhik 2000) for additional details) to obtain m!(m+δ) and using the result on a power series raised to a positive integer, we can write , where d 0,k+i−1 = c Applying the above equation and replacing f GW G (x) by the right hand side of (6), we obtain In other words, the pdf of the i th order statistic is exactly a linear combination of GWG densities with parameters (α, β, δ * ), where δ * = δi + δk + m.This is a very useful result since the properties of the order statistics of the GWG distribution can be obtained from those of EG distribution.For instance, we can obtain the moments of the i th order statistic from GWG distribution as: where E(Z l,j (x; α, β, δ * ) t ) is the t th moment of EG distribution with parameter β * * = β(δ * + i) + j, where δ * = δi + δk + m and ω i,j is given by (10).

Rényi entropy
In information theory, Rényi entropy generalizes Shannon entropy, Hartley entropy, minentropy, and collision entropy.Entropies quantify the diversity, uncertainty, or randomness of a system.Rényi entropy is named after Alfréd Rényi (Renyi 1961) and is very important in ecology and statistics as indices of diversity.Rényi entropy is also important in quantum information, where it can be used as a measure of entanglement.As an extension of Shannon entropy, Rényi entropy is a popular measure of entropy, which is defined by where ν > 0 and ν = 1.We obtain Rényi entropy for the GWG distribution as follows: The Gamma-Weibull-G Family of Distributions with Applications Using similar expansion for the pdf, that is, we obtain and b 0,m = 1, m = (βδ + 1)ν + βi.Now, we can write Rényi entropy as where

Estimation and observed information matrix
Estimation of the parameters of the GWG distributions can be obtained by the method of maximum likelihood.Let X ∼ GEM W (α, β, δ, θ) and ∆ = (α, β, δ, θ) T be the parameter vector.The log-likelihood for ∆ can be written as The first derivative of the log-likelihood function with respect to the parameters ∆ = (α, β, δ, θ) T are given by and where θ k is the k th element of the vector of parameters θ.
The partitioned observed information matrix ∆ = (α, β, δ, θ) for the GWG distribution is , which is symmetric matrix.We provide these elements as follows: and where , and θ k is the k th element of the vector of parameters θ.
The asymptotic confidence intervals for the parameters of the GWG distribution are as follows.
The expectations in the Fisher Information Matrix (FIM) can be obtained numerically.Let ∆ = (α, β, δ, θ) be the maximum likelihood estimate of ∆ = (α, β, δ, θ).Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, we have: , where I(∆) is the expected Fisher information matrix.The asymptotic behavior is still valid if I(∆) is replaced by the observed information matrix evaluated at ∆, that is J( ∆).The multivariate normal distribution N p (0, J( ∆) −1 ), where the mean vector 0 = (0, 0, 0, 0) T , can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions.That is, the approximate 100(1−η)% two-sided confidence intervals for α, β, δ, θ k are given by: respectively, where

Some sub-models of the GWU distribution
When α = β = δ = 1, the GWU distribution reduces to the exponential uniform (EU(θ), EU1) distribution with pdf as follows: By letting α = δ = 1, we obtain the Weibull uniform (WU(θ, β)) distribution as a special case of the GWU distribution, whose pdf is given by We can also obtain the gamma exponential uniform (GEU(θ, δ), GEU1) distribution by letting α = β = 1 with pdf given by When β = δ = 1, the GWU distribution reduces to the exponential uniform (EU(θ, α), EU2) distribution with pdf given by By setting δ = 1, we obtain the Weibull uniform distribution, which is also known as the Phani distribution (Phani 1987).The pdf of Phani distribution is given by Moreover, when α = 1, we can obtain the gamma Weibull uniform (GWU(θ, β, δ), GWU1) distribution as a special case of the GWU distribution, whose pdf is given by We can obtain the gamma exponential uniform (GWU(θ, α, δ), GU (Torabi and Montazeri 2012)) distribution by setting β = 1 and the corresponding pdf is given by When β = 2 and β = 2, δ = 1, we can obtain gamma Rayleigh uniform (GRU) distribution and Rayleigh uniform (RU) distribution, and the corresponding pdf are given by

Expansion of the GWU density
From equation ( 9), we can obtain where and denotes pdf of the exponentiated uniform (EU) distribution with power β(δ + i) + j.Thus, we can obtain statistical properties of the GWU distribution from those of EU distribution.

Hazard, reverse hazard and quantile functions
The hazard and reverse hazard functions of the GWU distribution are given by and respectively.In addition, we can also obtain the quantile function of GWU distribution from equation ( 14) and (15) as Several plots of pdf and hrf of the GWU distribution for selected parameter values are given by Figure 1 and 2. The graphs of the pdf take various types, including unimodal, decreasing, left and right skewed.As we can see from Figure 2, the plot for hrf of the GWU distribution contains decreasing, increasing, bathtub, unimodal and upside down bathtub shapes for the selected values of the GWU parameters.

Moments, incomplete moments, moment generating and characteristic functions
Let X ∼ GW U (α, β, δ, θ), 0 < x < θ, α, β, δ > 0, the s th moment of X can be obtained from (29) as Figure 1: Graphs of GWU pdf with selected parameters where E(Z s i,j ) denotes the s th moment of Z i,j , which follows the EU distribution with power β(δ + i) + j and ω i,j is defined as (30).Similarly, the incomplete moments and mgf are given by ω i,j E(e tZ i,j ), where E(e tZ i,j ) is the mgf of the EU distribution with power β * = β(δ + i) + j and ω i,j is defined as (30).The characteristic function is given by φ(t) = E(e itX ), where i = √ −1.Thus, we have where φ β(δ+i)+j (t) denotes the characteristic function of the EU distribution with power β(δ + i) + j, and ω i,j is defined as (30).
The first six moments, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS), and coefficient of kurtosis (CK) for selected values of the parameters of the GW U distribution are listed in Tables 2 below.The variance (σ 2 ), CV, CS and CK are given by respectively.The graphs of CS and CK versus δ show the dependence of skewness and kurtosis measures on the shape parameter.M oments (5.0,3.0,2.0,1.0)(1.0,1.0,1.0,1.0)(0.5,1.5,1.5,1.6)(1.0,1.0,1.5,1.5)Moreover, the conditional expectations of the GWU distribution can also be obtained from the EU (Cordeiro et al. 2013b) distribution with power β(δ + i) + j, and we present it as follow: where I i,j (y) = y 0 x s h β(δ+i)+j (x, θ)dx, and ω i,j is defined as (30).M edian(X) the median.Then the mean deviation about the mean and the mean deviation about the median can be obtained from

Mean deviation, Bonferroni and Lorenz curves
respectively, where m(z) = z 0 xf GW U (x)dx is the first incomplete moment.Bonferroni and Lorenz curves are given by respectively, where I β(δ+i)+j (i, j) = q 0 xf β(δ+i)+j (x, θ)dx, is the first incomplete moment of the EU distribution with power β(δ + i) + j and ω i,j is defined as (30).

Order statistics
The distribution of the i th order statistic for the GWU distribution is presented in this section.Let X 1 , . . ., X n be i.i.d random variables distributed according to (29).The pdf of the i th order statistic, for example X i:n , is given by that is, the pdf of the i th order statistic is a linear combination of GWU densities with parameters (α, β, δ * , θ), where δ * = δi + δq + m.This is a very useful result since properties of the order statistics of the GWU distribution can be obtained from those of EU distribution.
Graphs of the GWW density functions with several combinations of parameter values are given by Figure 4, which shows different shapes including negative skew, positive skew, monotonically decreasing and almost symmetric bell-shape.

Some sub-models of the GWW distribution
When α = β = δ = k = 1, the GWW distribution reduces to the exponential exponential (EE) distribution with pdf as follows: By letting α = β = δ = 1, we obtain the exponential power (EP) distribution (Smith and Bain 1975) as a special case of the GWW distribution, whose pdf is given by We can also obtain Chen distribution (Chen 2000) by letting β = δ = λ = 1 with pdf given by When β = δ = k = 1, the GWW distribution reduces to Gompertz distribution (Gompertz 1895) with pdf given by By setting α = β = 1, we obtain the gamma exponential power (GEP) distribution.The pdf of GEP distribution is given by Moreover, when α = δ = 1, we obtain the exponentiated exponential power (EEP) distribution as a special case of the GWW distribution, whose pdf is given by We also obtain the gamma exponentiated exponential power (GEEP) distribution by setting α = 1 and the corresponding pdf is given by Figure 4: Graphs of GWW pdf with selected parameters.Note that, the focus and originality of this paper is that the GWG family of distributions are capable of generating bathtubshaped probability density functions (pdf).Because of its importance and in a different scale, we are presenting a bathtub-shaped pdf in a separated plot (right panel) as above.

Expansion of the GWW density
From equation ( 9), we can obtain where and denotes pdf of the exponentiated Weibull (EW) (Nassar and Eissa 2003) distribution with power β(δ+i)+j.Therefore, we can obtain the statistical properties of the GWW distribution from those of the EW distribution.

Hazard, reverse hazard and quantile functions
The hazard and reverse hazard functions of the GWW distribution can be obtained as respectively.The graphs of the hrf for selected parameters are given in Figure 5.The plots show various shapes including monotonically decreasing, monotonically increasing, and bathtub shapes for several combinations of parameters values.This attractive flexibility makes the GWW hazard function be useful and suitable for monotonic and non-monotonic empirical hazard behaviors which are more likely to be encountered in real life situations.
In addition, we can also obtain the quantile function of GWW distribution from equation ( 14) and ( 15) as

Moments, incomplete moments, moment generating and characteristic functions
Let X ∼ GW W (α, β, δ, λ, k), x ≥ 0, α, β, δ, λ, k > 0, the s th moment of X can be obtained from ( 35) as where E(Z s i,j ) denotes the s th moment of Z i,j , which follows the EW distribution (Nassar and Eissa 2003), with power β(δ + i) + j and ω i,j is defined as (36).The s th moment of Z i,j is given by (Pal, Ali, and Woo 2006) and we present it as follows: Similarly, the incomplete moments and mgf can be obtained as where I i,j (y) = y 0 x s h β(δ+i)+j (x, θ)dx and M X (t) = ∞ i,j=0 ω i,j E(e tZ i,j ), where E(e tZ i,j ) is the mgf of the EW (Cordeiro et al. 2013b) distribution with power β(δ + i) + j and ω i,j is defined as (36).The characteristic function is given by φ(t) = E(e itX ), where i = √ −1.Thus, the characteristic function is given by where φ β(δ+i)+j (t) denotes the characteristic function of the EW distribution with power β(δ + i) + j and ω i,j is defined as (36).
In addition, the conditional expectations of the GWW distribution can also be obtained from the EW (Nassar and Eissa 2003) distribution as follow: where I i,j (y) = y 0 x t h β(δ+i)+j (x, θ)dx is the t th incomplete moment of EW and ω i,j is defined as (36).

Order statistics
The distribution of the i th order statistic for the GWW distribution is presented in this section.Let X 1 , . . ., X n be i.i.d random variables distributed according to (35).The pdf of the i th order statistics, for example X i:n , is given by that is, the pdf of the i th order statistic is a linear combination of GWW densities with parameters (α, β, δ * , λ, k), where δ * = δi + δq + m.
Therefore, properties of the order statistics of the GWW distribution can be obtained from those of EW (Nassar and Eissa 2003) distribution.For instance, we can obtain the the t th moment of the i th order statistics and is given by where E(Z l,j (x; α, β, δ * ) t ) is the t th moment of EW distribution with power β(δ * + i) + j, where δ * = δi + δq + m and ω i,j is defined as (36).

Monte Carlo simulation study
In this section, a simulation study is conducted to assess the performance and examine the mean estimate, average bias, root mean square error of the maximum likelihood estimators for each parameter for both GWU and GWW distributions.We study the performance of the GW U and GW W distributions by conducting various simulations for different parameter values.The simulation study is repeated for N = 5000 times each with sample size n = 25, 50, 75, 100, 200, 400, 800, 1600, 3200 and parameter values 2, δ = 0.8, θ = 1.5, for the GWU distribution and I : α = 0.8, β = 1.6, δ = 0.3, λ = 0.4, k = 1, II : α = 0.9, β = 1.2, δ = 0.5, λ = 0.6, k = 1.2, III : α = 1.0, β = 1.0, δ = 0.8, λ = 1.0, k = 1.5, IV : α = 0.5, β = 0.8, δ = 0.5, λ = 0.8, k = 0.5, for the GWW distribution, respectively.Tables 4, 5 and 6, 7 lists the means MLEs of the model parameters along with the respective average bias and root mean squared errors (RMSE) for the GWU and GWW distributions, respectively.From the results, we can verify that as the sample size n increases, the mean estimates of the parameters tend to be closer to the true parameter values, since RMSEs decay toward zero.The bias and RMSEs are given by: respectively.

Applications
In this section, we present two examples to illustrate the flexibility of the GWU and GWW distributions and their sub-models for data modeling.
The first data consists of the lifetimes of n = 50 devices given by Aarset data (Aarset 1987).
Some descriptive statistics of these two data sets are given in Table 8.
The maximum likelihood estimates (MLEs) of the GWU and GWW parameters α, β, δ, and θ are computed by maximizing the objective function via the subroutine NLMIXED in SAS.The estimated values of the parameters (standard error in parenthesis), -2log-likelihood statistic, Akaike Information Criterion, AIC = 2p − 2 ln(L), Bayesian Information Criterion, BIC = p ln(n)−2 ln(L), and Consistent Akaike Information Criterion, AICC = AIC +2 p(p+1) n−p−1 , where L = L( ∆) is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, and p is the number of estimated parameters, and Kolmogorov- ) are presented in Table 9 and 10.In order to compare the models, we use the criteria stated above.Note that for the value of the log-likelihood function at its maximum ( n ), larger value is good and preferred,    and for the Kolmogorov-Smirnov test statistic (K-S), smaller value is preferred.The GWU distribution is fitted to the Aarset data (Aarset 1987) set and these fits are compared to the fits using the EU(θ), WU(θ, β), GEU(θ, δ), EU(θ, α), Phani, GWU(θ, β, δ), GEU(θ, α, δ) and GWU distributions.Similarly, the GWW distribution is also fitted to the Meeker and Escobar (Meeker and Escobar 1998) data set and these fits are compared to the fits using the EE, EP, Chen, Gompertz, GEP, EEP and GEEP distributions.The GWW and GWU distributions was compared with the non-nested beta exponentiated Weibull (BEW) (Cordeiro, Gomes, da Silva, and Ortega 2013a), and beta Weibull (BW) distributions as well.The pdf of the BEW distribution is given by When α = 1, we have the BW distribution.We also compared the GWW and GWU distributions with the gamma generalized modified Weibull (GGMW) (Oluyede, Huang, and Yang (2015)) and gamma modified Weibull (GMW) distributions, respectively.The pdf of the GGMW distribution (Oluyede et al. 2015) is given by [− log(1 − e −αx−βx θ e λx )] δ−1 (α + βx θ−1 e λx [θ + λx])e −αx−βx θ e λx .(39) When α = 0, we have the GMW distribution.
The LR test rejects the null hypothesis if ω > χ 2 , where χ 2 denote the upper 100 % point of the χ 2 distribution with 2 degrees of freedom.
We also computed a measure of closeness of each plot to the diagonal line.This measure of closeness (Chambers, Cleveland, Kleiner, and Tukey 1983) is given by the sum of squares (SS) G GW G (x (j) ; α, β, δ, θ) − j − 0.375 n + 0.25 2 , which is also listed in Table 9 and Table 10.
The 95% confidence intervals for the GWU model (Aarset Data)   For the Aarset data, the LR test statistic of the hypothesis H 0 : Phani against H a : GWU and H 0 : GEU(θ, α, δ) against H a : GWU are ω 1 = 12.00 with p-value= 5.32 × 10 −4 and ω 2 = 17.5 with p-value= 2.87×10 −5 .Therefore, we reject H 0 in favor of H a and conclude that the GWU distribution is significant better than the WU and GEPU distributions.Moreover, the values of the statistics AIC, AICC and BIC show that model GWU is a "better" fit for this data.The value of the goodness-of-fit statistic KS is smallest for the GWU distribution.Also, the value of SS for GWU given by the probability plot is smallest for this model.Consequently, the GWU distribution is a "better"fit when compared to the nested distributions and non-nested four parameter GMW and BW distributions.
Similarly, we can also conduct the LR test for the Meeker and Escobar data for the hypothesis H 0 : GEP against H a : GWW and H 0 : GEEP against H a : GWW.The LR test statistic of these hypothesis are ω 3 = 24.8 with p-value= 4.12 × 10 −6 and ω 4 = 13.7 with p-value= 2.14 × 10 −4 , which implies that we should reject H 0 in favor of H a and conclude that the GWW distribution is a significant better fit for the Meeker and Escobar data.In addition, the values of the statistics AIC, AICC and BIC clearly show that model GWW is a "better"fit for this data.
and Z η 2 is the upper η 2 th percentile of a standard normal distribution.

Figure 2 :
Figure 2: Graphs of GWU hrf with selected parameters

Figure 5 :
Figure5: Graphs of GWW hazard rate functions (hrf) with selected parameters.Similar with pdf of GWW, we are presenting a bathtub-shaped hrf separately in the right panel.

Figure 7 :
Figure 7: Fitted pdf and observed probabilities of the GWU distribution for Aarset data

Table 4 :
Monte Carlo simulation results for GWU distribution: mean estimate, average bias, and RMSE

Table 5 :
Monte Carlo simulation results for GWU distribution: mean estimate, average bias, and RMSE

Table 7 :
Monte Carlo simulation results for GWW distribution: mean estimate, average bias, and RMSE

Table 8 :
Descriptive statistics of application data sets

Table 9 :
GWU estimation for Aarset data