The Gamma Odd Weibull Generalized-G Family of Distributions: Properties and Applications

A new generalized family of models called the Gamma Odd Weibull Generalized-G (GOWG-G) family of distributions is proposed and studied. Properties of the new family of distributions including moments, conditional moments, distribution of the order statistics and RØnyi entropy are derived. Maximum likelihood estimation technique is used to estimate the model parameters. Four special cases of the GOWG-G family of distributions are considered. A simulation study was carried out to examine the accuracy of the Maximum Likelihood Estimates (MLE) of the parameters.


Introduction
There are several extensions and modications of the Weibull distribution in the literature meant to accomodate both monotone and non-monotone hazard rate functions.Extended and generalized distributions for use in the modeling of monotone and non-monotone hazard rate functions, particularly in reliability and survival analysis are very crucial.Generalized distributions including modied Weibull distributions (see Oluyede et al. (2015) for details) play very prominent role in the modelling of monotone and non-monotone hazard rate functions, exploring tail variation and improved goodness-of-t tests.
Over the years, several new and very useful families of distributions have been developed and new generalized distributions obtained by adding one or more parameters to existing distributions in the statistical literature including the Transform-transformer (T-X) by Alzaghal et al. (2013), Weibull-G by Bourguignon et al. (2014), beta-G by Eugene et al. (2002), McDonald-G (Mc-G) by Alexander et al. (2012) and Lomax generator by Cordeiro et al. (2014), Kumaraswamy odd log-logistic family by Alizadeh, Emadi, Doostparast, Cordeiro, Ortega & Pescim (2015), Kumaraswamy Marshall-Olkin family by Alizadeh, Tahir, Cordeiro, Mansoor, Zubair & Hamedani (2015).Peter et al. (2021) developed and studied the gamma odd Burr III family of distributions.Cordeiro et al. (2013) introduced a class of distributions called the exponentiated generalized (EG) class of distributions.The process of nding an adequate model to make inferences is a very important problem in statistical modeling.Barreto-Souza et al. (2013) stated that adding parameters to an established baseline distribution is a well established technique for obtaining more exible new families of distributions.
The generalized-G (G-G) class of distributions extended several well know distributions including distributions in the literature such as the exponential, Weibull, gamma, Fréchet and Gumbel distributions.An attractive feature about the model is that the extra parameter that is introduced can control both tail weight and possibly adding entropy to the density function, depending on the resulting distribution.
Let X be a random variable with cumulative distribution function (cdf) G(x; ξ) depending on the parameter vector ξ.The cdf and probability density function (pdf) of the generalized-G (G-G) are given by for α > 0, x ∈ R. The corresponding hazard rate function (hrf) is given by (2) Alzaghal et al. (2013) dened the T − X family of distributions given by where W (G(x)) satises the following conditions: (1) ) is diierentiable and monotonically non-decreasing, and (3) W (G(x)) → a as x → −∞ and W (G(x)) → b as x → ∞.In this note, we consider the transformation W (G(x; ξ)) = 1−G α (x;ξ) G α (x;ξ) for the baseline cumulative distribution function (cdf) G(x; ξ).In fact this paper is concerned with the development and extension of the gamma operator via the odd Weibull generalized-G transformation by introducing additional shape parameters to obtain a new generalized distribution called the gamma odd Weibull generalized-G (GOWG-G) family of distributions.A comparison of special cases of the new members of the family of distributions with several other distributions in the literature via several goodness-of-t statistics is conducted in order to establish the eectiveness and usefulness of this new distribution.
The new distribution allows for modeling of asymmetric, heavy and long tailed, left skewed, reverse-J shapes, as well for special models that provide better ts than the baseline distribution.The general motivations for the construction and development of GOWG-G family of distributions in practice are: to generalize the odd Weibull-G and odd Weibull generalized-G families of distributions; to develop distributions that can readily deal with or handle monotone and non-monotone hazard rate functions; to obtain very exible distributions that take into consideration not only shape but also skewness, kurtosis and tail variation, as well as to improve goodness-of-t to real data with wider applications in several areas including environmental sciences, reliability, and actuarial sciences.
The results in this note are organized in the following manner.Section 2 contain the new GOWG-G family of distributions and its sub-models, hazard rate function and the quantile function.In Section 3, moments and generating function, conditional and incomplete moments are presented.In Section 4, the distribution of order statistics and Rényi entropy are presented.Section 5 contain the estimation of the parameters of the GOWG-G family of distributions via the method of maximum likelihood, followed by a Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimators in Section 6.Some applications to real data sets are given in Section 7, followed by concluding remarks in Section 8, and future work in Section 8.The derivation of some of the statistical properties of the gamma odd Weibull generalized-G (GOWG-G) family of distributions including sub-models, expansion of the density, hazard rate function, and quantile function are presented in this section.

The Model
Let us rst consider the cdf and pdf of the odd Weibull generalized-G (OWG-G) family of distributions given by respectively, for α, β > 0 and parameter vector ξ.If a random variable X has the OWG-G family of distributions, we write X ∼ OW G − G(α, β, ξ).Note that if T is a random variable denoting the lifetime of a system with distribution function K(x; ξ) = 1 − G α (x; ξ) and the random variable X is the odds ratio, then the risk that the system with lifetime T will fail at time x is given by K(x;ξ) . Now, suppose we model the randomness of the odds ratio via the Weibull distribution with shape parameter β > 0, that is, r(t; β) = βt β−1 e −t β , for t > 0, then the cdf of the random variable X is the new OW G − G(β, α, ξ), that is, Note that when α = 1, we obtain the odd Weibull-G (OW-G) family of distributions (Bourguignon et al., 2014) with the pdf for β > 0, and parameter vector ξ.
The cdf and pdf of the proposed gamma odd Weibull generalized-G (GOWG-G) family of distributions are given by Revista Colombiana de Estadística -Applied Statistics 46 (2023) 144 and respectively, for α, β, δ > 0 and parameter vector ξ.If a random variable X has the GOWG-G family of distributions, we write X ∼ GOW G − G(β, α, δ, ξ).

Sub-models of the GOWG-G Family of Distributions
In this subsection, some sub-families of the GOWG-G family of distributions are presented.
When δ = 1, we obtain the odd Weibull generalized-G (OWG-G) family of distributions with pdf for α, β > 0, and parameter vector ξ.This is a new family of distributions.
When β = 1, we obtain the gamma odd exponential generalized-G (GOEG-G) family of distributions wth pdf for α, δ > 0 and parameter vector ξ.This is a new family of distributions.
When β = 2, we obtain the gamma odd Rayleigh generalized-G (GORG-G) family of distributions with the pdf for α, δ > 0 and parameter vector ξ.This is a new family of distributions.
When α = β = 1, we obtain the gamma odd exponential-G (GOE-G) family of distributions with the pdf for δ > 0 and parameter vector ξ.This is a new family of distributions.
If α = δ = 1, and β = 1, we obtain the odd exponential-G (OE-G) family of distributions with the pdf for the parameter vector ξ.
When α = δ = 1, and β = 2, we obtain the odd Rayleigh-G (OR-G) family of distributions with the pdf for the parameter vector ξ.
When α = 1 and β = 2, we have gamma odd Rayleigh-G (GOR-G) family of distributions with pdf for δ > 0 and parameter vector ξ.
If δ = 1, and β = 2, we have the odd Rayleigh generalized-G (ORG-G) family of distributions with pdf for α > 0, and parameter ξ.This is a new family of distributions.If α = 1, we obtain the gamma odd Weibull-G (GOW-G) family of distributions with pdf for β, δ > 0 and parameter vector ξ.This is a new family of distributions.
We also note that dierent baseline cdf G(x; ξ) lead to several new gamma generalized distributions as sub-models.Furthermore, a number of sub-models are possible by changing the parameter vector ξ, and these special cases as well as corresponding parameters are presented in Table 1.

GOWG-Kumaraswamy
Modied Exponential GOWG-Exponential 2.3.Hazard Rate and Quantile Functions In this section, we present the hazard rate and quantile functions of the GOWG-G family of distributions.The hazard rate function (hrf) of the GOWG-G family of distributions is given by The quantile function of the GOWG-G family of distributions is obtained by solving the non-linear equation: Equivalently, Consequently, the quantile function for the GOWG-G family of distributions is given by (21) It follows therefore that random numbers can be generated from the GOWG-G family of distributions based on equation ( 21), for specied baseline cdf G.

Some Special Cases
In this section, we consider some special cases of the GOWG-G family of distributions, specically when the baseline distribution function G(x; ξ) are uniform, log-logistic, logistic and Weibull distributions, respectively.There are several new sub-models that can be readily obtained from these special cases for selected values of the model parameters.Plots of the density functions as well as the hazard rate functions for these special cases are presented.

GOWG-Uniform Distribution
Suppose the cdf and pdf of the baseline distribution are given by G(x; θ) = x θ and g(x; θ) = 1 θ for 0 < x < θ < ∞.Then, the new GOWG-Uniform distribution has the cdf and pdf given by and Figure 1 represents the plots of density function and hrf of the GOWG-Uniform distribution for several combinations of the parameters α, β, δ, and θ.
The pdfs of GOWG-Uniform distribution takes on various shapes, including uni-modal, right-skewed and reverse-J.Further, hazard rate functions of GOWG-Uniform distribution exhibit decreasing, increasing, bathtub and upside-down bathtub followed by bathtub shapes.

GOWG-Log-logistic Distribution
Suppose the cdf and pdf of the baseline distribution are given by G( The new GOWG-Log-logistic distribution has cdf and pdf given by Figure 1: Plots of the pdf and hrf for the GOWG-Uniform distribution and for α, β, δ, c > 0, and x > 0. The GOWG-Log-logistic survival function and hrf are given by and Figure 2 illustrates the graphs of the density function and hrf of the GOWG-Log-logistic distribution using several combinations of the parameters α, β, δ, and c.

Figure 2: Plots of the pdf and hrf for the GOWG-Log-logistic distribution
In the GOWG-Log-logistic distribution, the pdf appears to be in various shapes, such as unimodal, reverse-J and left-or right-skewed.The plots of the GOWG-Loglogistic distribution hrf shows decreasing, increasing, bathtub and upside-down bathtub shapes.

GOWG-Logistic Distribution
Suppose the cdf and pdf of the baseline distribution are given by G( Then the new GOWG-Logistic distribution has cdf and pdf given by and for α, β, δ, σ > 0. The survival function and hrf are given by and Figure 3 shows graphs of density function and hrf for dierent combinations of GOWG-Logistic parameters.
It appears that the pdf of the GOWG-Logistic distribution can take various shapes, such as unimodal, increasing and left-or right-skewed.Plots of the GOWG-Logistic distribution hrf show decreasing, increasing, upside-down bathtub and upside-down bathtub followed by bathtub shapes.

GOWG-Weibull Distribution
Suppose the cdf andf pdf of the baseline distribution are given by G(x; c) = 1 − exp(−x c ) and g(x; c) = cx c−1 exp(−x c ) for c > 0 and x > 0. Then the new GOWG-Weibull distribution has cdf and pdf given by and and for α, β, δ, c > 0, and x > 0. Figure 4 illustrates the graphs of the density function and hrf of the GOWG-Weibull distribution using several combinations of the parameters α, β, δ, and c.It appears that the pdf of the GOWG-Weibull distribution can take various shapes, such as unimodal, reverse-J and left-or right-skewed.Plots of the GOWG-Weibull distribution hrf show decreasing, increasing, and bathtub shapes.

Series Expansion of Density Function
In this section, we present the series expansion of the GOWG-G family of density functions.
and applying the result on power series raised to a positive integer, with where b s,m = (sa 0 ) −1 s l=1 [m(l + 1) − s]a l b s−l,m , and b 0,m = a m 0 , (Gradshteyn & Ryzhik, 2000), and applying the exponential series expansion, as well as the generalized binomial theorem: and we have where g * q+t+1 (x; ξ) = (q + t + 1)[G(x; ξ)] q+t+1−1 g(x; ξ) is the exponentiated-G (exp-G) pdf with the exponentiation parameter q + t + 1 > 0 and parameter vector ξ, where ω(q, t) is given by Thus, the pdf of the GOWG-G family of distributions is a mixture of exp-G densities.Consequently, the mathematical and statistical properties of the GOWG-G family of distributions follow directly from those of the exp-G distribution.Please see appendix A for details of the series expansion.
Note that using the fact that Broderick Oluyede & Gayan Warahena-Liyanage the cdf of the GOWG-G family of distributions can be written as where and ω(q, t) is given by equation ( 42).The corresponding GOWG-G pdf is given by

Moments, Conditional and Incomplete Moments
In this section, moments, moment generating function, conditional moments for the GOWG-G family of distributions are presented.Moments are very useful in the study of important features and characteristics of a distribution (e.g., central tendency, dispersion, skewness and kurtosis).These measures (moments, moment generating function, incomplete moments) can be readily obtained for the submodels given in section 2.

Moments and Generating Function
Let Y k+1 ∼ Exponentiated − G(k + 1, ξ), then the r th raw moment, µ ′ r of the GOWG-G family of distributions is given by: The moment generating function (MGF), for |a| < 1, is given by: The coecients of variation (CV), Skewness (CS) and Kurtosis (CK) can be readily obtained.The variance (σ 2 ), standard deviation (SD=σ), coecient of variation (CV), coecient of skewness (CS) and coecient of kurtosis (CK) can be readily obtained for specied baseline distribution.Also, note that the r th cumulant of the random variable X can be readily obtained from the recursive relationship: r , so that the CS and CK are given by γ We present in Figures 5 and 6, 3D plots of skewness and kurtosis of the GOWG-Log-logistic distribution.For lifetime models and measures of inequality, it is of particular interest to nd the conditional and incomplete moments.The r th conditional moments of the GOWG-G family of distributions is given by Revista Colombiana de Estadística -Applied Statistics 46 (2023) 144 where for α, β, δ > 0, and parameter vector ξ.The mean residual life function is given by E(X − a|X > a) = E(X|X > a) − a = V F (a) − a, where V F (a) is referred to as the vitalilty function of the distribution function F. The mean deviations, Bonferroni and Lorenz curves can be readily obtained from the conditional and incomplete moments.

Order Statistics and Rényi Entropy
Order statistics and entropy play important roles in probability and statistics, particularly in reliability, lifetime data analysis and information theory.In this section, we present the distribution of the i th order statistics and Rényi entropy for the GOWG-G family of distributions.

Order Statistics
In this subsection, the pdf of the i th order statistic is presented.Let X 1 , X 2 , . .., X n be independent and identically distributed GOWG-G random variables.Using the binomial expansion the pdf of the i th order statistic from the GOWG-G pdf f (x; α, β, δ, ξ) = f (x) can be written as where f m+i (x; α, β, δ, ξ) is the exponentiated gamma odd generalized-G pdf with exponentiated parameter m + i > 0 and weights ω(i, m) given by Note that from the results given by Hosseini et al. (2018), we can also write the pdf of the i th order statistic as where g * r+k+1 (x) is the exp-G density function with power parameter r and C r is dened by the equation ( 45).The quantities f j+i−1,k are given by the recursively by We can also obtain the distribution of the i th order statistic as follows, by applying the result of Gradshteyn & Ryzhik (2000) of a power series raised to a positive integer: where d j+i−1,0 = C j+i−1 0 , and for k ≥ 1, Replacing the GOWG-G pdf f (x) by equation ( 46) and applying the result on product of two series (Gradshteyn & Ryzhik, 2000), we have where b r = C r+1 (r + 1), and Consequently, the pdf of the i th order statistic from the GOWG-G family of distributions can be written as Revista Colombiana de Estadística -Applied Statistics 46 (2023) 144 where and g * k+1 (x; ξ) = (k + 1)(G(x; ξ)) k g(x; ξ) is the exp-G density function with power parameter k+1.Thus, the pdf of the GOWG-G distribution is an innite mixture of exp-G densities.The structural properties such as moments, incomplete moments of the distribution of the i th order statistic and other measures of the GOWG-G distribution follows or can be readily obtained from those of the exp-G distribution.

Maximum Likelihood Estimation
Let X ∼ GOW G − G(α, β, δ, ξ) and ∆ = (α, β, δ, ξ) T be the vector of model parameters.The log-likelihood function ℓ n = ℓ n (∆) based on a random sample of size n from the GOWG-G family of distributions is given by The elements of the score vector U (∆) are given in appendix B. The maximum likelihood estimates of the parameters, denoted by ∆ is obtained by solving the nonlinear equation ( ∂ℓn ∂α , ∂ℓn ∂β , ∂ℓn ∂δ , ∂ℓn ∂ξ k ) T = 0, using a numerical method such as Newton-Raphson procedure.The multivariate normal distribution N q+3 (0, J( ∆) −1 ), where the mean vector 0 = (0, 0, 0, 0) T and J( ∆) −1 is the observed Fisher information matrix evaluated at ∆, can be used to construct condence intervals and condence regions for the individual model parameters and for the survival and hazard rate functions.

Simulation Study
The performance of the GOWG-LLoG is examined by conducting various simulations for dierent sizes (n=25, 50, 75, 100, 200, 400, 800, 1200) via the R package.We simulate N = 1000 samples for the true parameters values given in Table 2.Additional simulation results are available upon request or in the appendix C. The tables list the mean MLEs of the model parameters along with the respective average bias and root mean squared errors (RMSEs).The average bias and RMSE for the estimated parameter, say, θ, say, are given by: respectively.As we can see from the results, RMSE decreases as the sample size n increases, so the mean estimates of parameter values are closer to the true parameter values.In this section, we present examples to illustrate the exibility and usefulness of the GOWG-LLoG and GOWG-W distributions for data modeling.The GOWG-LLoG distrbution is tted to the data set in subsection 7.1 while the GOWG-W distribution is tted to the data sets in subsections 7.2 and 7.3.GOWG-LLoG tted in subsection 7.1, and GOWG-W in subsections 7.2 and 7.3.These ts are compared to the ts of several competing non-nested distributions with equal number of parameters.The GOWG-LLoG distribution is compared with exponentiated power generalized Weibull (EPGW) (Péna-Ramirez et al., 2018), burr XII Poisson (BXIIP) (da Silva et al., 2015), Topp-Leone-Marshall-Olkin-Weibull (TLMO-W) (Chipepa et al., 2020), beta Weibull (BW) (Lee et al., 2007), the exponential Lindley odd loglogistic Weibull (ELOLLW) (Korkmaz et al., 2018) and Kumaraswamy-Weibull (KwW) (Cordeiro et al., 2010).
In Figure 12, we see that the cdf line for the GOWG-W distribution indicated by the blue line is closer to the empirical cdf while the survival function in blue is also close to the Kaplan-Meier(K-M) curve which indicate that our model is the best in explaining the aircraft windshield data.The TTT plot for aircraft windshield data indicates a increasing hazard rate function, hence the aircraft windshield data can be tted to our model.The third data set refers to the level of mercury in 34 albacore caught in the Eastern Mediterranean obtained from Mol et al. (2012).The observations are as follows: 1. 007, 1.447, 0.763, 2.010, 1.346, 1.243, 1.586, 0.821, 1.735, 1.396, 1.109, 0.993, 2.007, 1.373, 2.242, 1.647, 1.350, 0.948, 1.501, 1.907, 1.952, 0.996, 1.433, 0.866, 1.049, 1.665, 2.139, 0.534, 1.027, 1.678, 1.214, 0.905, 1.525, 0.763  Table 5 indicates that GOWG-W has the highest p-value for the K-S statistic and the lowest values of all goodness-of-t statistics.The GOWG-W model therefore works better with level of mercury data than EPGW, BGL, BW, KOL-LLoG, NMW, and OLLEW models.Moreover, Figure 13 shows that our model has the lowest SS value from the probability plots compared to the competing non-nested models on the level of mercury data.
In Figure 14, we see that the cdf line for the GOWG-W distribution indicated by the blue line is closer to the empirical cdf while the survival function in blue is also close to the Kaplan-Meier(K-M) curve which indicate that our model is the best in explaining the level of mercury data.The TTT plot for level of mercury data indicates a increasing hazard rate function, hence the level of mercury data can be tted to our model.

Concluding Remarks
A new generalized distribution called the gamma odd Weibull generalized-G (GOWG-G) family of distributions is developed and presented.The GOWG-G distribution has several new and known distributions as special cases or sub-models.The behaviour of the hazard rate function of the GOWG-G family of distributions is exible.We also obtain closed form expressions for the moments, mean and median deviations, distribution of order statistics and entropy.Maximum likelihood estimation technique is used to estimate the model parameters.The performance of the special case of the GOWG-G family of distributions was examined by conducting various simulations for dierent sizes.Finally, two special cases of the GOWG-G family of distributions are tted to real data sets to illustrate the applicability and usefulness of the distributions.

Future Work
We hope that the new GOWG-G family of distributions will contribute valuable information toward generalizing odd-Weibull-G as well as odd-Weibull generalized-G families of distributions in terms of exibility and versatility.Moreover, from the standpoint of practical applications, we believe that we should estimate the pa-  Afy, Nassar, Cordeiro & Kumar (2020).In a future paper, we will continue investigating this aspect, and we hope that our study will serve as a reference for future research in this area.41) (q + t + 1)k! αβ Γ(δ) (q + t + 1)[G(x; ξ)] q+t g(x; ξ).

Figure 3 :
Figure 3: Plots of the pdf and hrf for the GOWG-Logistic distribution

Figure 4 :
Figure 4: Plots of the pdf and hrf for the GOWG-Weibull distribution Figure 5: 3D plots of the skewness for the GOWG-Log-logistic distribution for some selected parameter values.

.
where x (j) are the ordered values of the observed data.The measures of closeness are given by the sum of squares SS = n j=1 F GOW G−G (x (j) ; α, β, δ, ĉ) − j − 0.375 n + 0.25 2 These plots are shown in Figures 9, 11 and 13.Revista Colombiana de Estadística -Applied Statistics 46 (2023) 144

Figure 9 :
Figure 9: Histogram, tted density and probability plots for failure times data

Figure 11 :
Figure 11: Histogram, tted density and probability plots for aircraft windshield data

Figure 12 :
Figure 12: Estimated cdf, Kaplan-Meier survival and scaled TTT-Transform plots for the GOWG-W distribution for aircraft windshield data

Figure 13 :
Figure 13: Histogram, tted density and probability plots for level of mercury data.

Figure 14 :
Figure 14: Estimated cdf, Kaplan-Meier survival and scaled TTT-Transform plots for the GOWG-W distribution for level of mercury data.

Table 2 :
Monte Carlo simulation results for GOWG-LLoG distribution: mean, average bias and RMSE.

Table 3 :
Parameter estimates and goodness-of-t statistics for various models tted for failure times data.

Table 4 :
Parameter estimates and goodness-of-t statistics for various models tted for aircraft windshield data.
. The estimated variance-covariance matrix for GOWG-W model on level of mercury data set is given by

Table 5 :
Parameter estimates and goodness-of-t statistics for various models tted for level of mercury data.