The Gamma Odd Burr X-G Family of Distributions with Applications

A new family of distributions called Gamma Odd Burr X-G (GOBX-G) distribution is introduced in this paper. Its structural properties such as the survival function, hazard function, density expansion, quantile function, moments and generating functions, incomplete moments, probability weighted moments, R ´ enyi entropy and order statistics were derived. Maximum likelihood technique is used to estimate the parameters of this model. The ﬂexibility and applicability of this model is demonstrated using real life datasets.


Introduction
Lifetime models are very useful for describing and predicting real world phenomena found in medicine, sciences, economics and many other areas. The usefulness of these models is usually seen in products lifetime evaluations such as in reliability and survival analysis. Numerous studies have been conducted to find suitable lifetime models that best describes the real world phenomena. For the past decade, researches mainly focused in developing new family of distributions through addition of extra parameter/s in the traditional and common families of continuous distributions. These added parameters have shown to be producing much flexible models that fits real world datasets better than traditional models (Barreto-Souza et al.  Lahcene (2021) also developed an extended-Gamma family of distributions. Additionally, Burr X distribution was also extended to Exponentiated Generalized Burr X distribution (Khaleel et al., 2018), Weibull Burr X distribution (Ibrahim et al., 2017), Exponentiated Burr X distribution (Ahmed et al., 2021), Beta Burr X distribution , Power Burr Type X distribution (Usman and Ilyas, 2020) and Type I Half-Logistic Burr X distribution (Shrahili et al., 2019). Khaleel et al. (2016) also developed a Gamma Burr type X distribution via the Gamma generator and a two parameter Burr X distribution introduced by Surles In equation (1) and (2), if we let G(x) = F(x; θ, ξ) and g(x) = f(x; θ, ξ), we get a new family of distributions called The Gamma Odd Burr X-G Family of Distributions with Applications the Gamma Odd Burr X-G (GOBX-G) family of distributions with the cdf and pdf given as and f(x; δ, θ, ξ) = 2θg(x; ξ)G(x; ξ) respectively, for x, δ, θ > 0 and ξ being a vector of parameters from the baseline distribution, where γ (δ, z) = z 0 t δ−1 e −t dt is the lower incomplete gamma function. The survival function of the GOBX-G family of distributions is given as The hazard rate function (hrf) and the cumulative hazard function (chf) of GOBX-G family of distributions are respectively given by

Series Expansion of the GOBX-G Density Function
A series expansion of the GOBX-G density follows in this section. If we let y = exp − G(x;ξ) , then we can write equation (4) as f(x; δ, θ, ξ) = 2θ δ g(x; ξ)G(x; ξ) The Gamma Odd Burr X-G Family of Distributions with Applications Using series representation −log(1 − y) = ∞ i=0 y i+1 i+1 , for y ∈ (0, 1), we have Thus, we can write the GOBX-G density as Next, let a s = (s + 2) −1 and considering the result of a power series raised to a positive integer we get Grandshteyn and Ryzhik (1980)). The GOBX-G density function then simplifies to Applying the generalized binomial series representation (1 − z) n = ∞ i=0 (−1) i n i z i which is valid for |z| < 1, we can express the GOBX-G density as Applying the exponential series representation exp(−z) = ∞ k=0 , the density further expand to The Gamma Odd Burr X-G Family of Distributions with Applications Applying the generalized binomial series representation in G 2k+1 (x; ξ) = 1 − G(x; ξ) 2k+1 , we get The GOBX-G density function can finally be expressed as an infinite linear combination of Exponentiated G (Exp-G) density functions as where and h p+1 (x; ξ) = (p + 1)g(x; ξ)G p+1−1 (x; ξ) is the Exponentiated-G pdf with power parameter p + 1. Thus, mathematical properties of GOBX-G family of distributions are evident from those of the Exp-G distributions.

Quantile Function
This section present the quantile function for the GOBX-G distribution. The quantile function can be derived from the cdf of GOBX-G distribution in equation (3) by solving the non-linear equation This simplifies to The Gamma Odd Burr X-G Family of Distributions with Applications Replacing G(x; ξ) with 1 − G(x; ξ), we get which finally gives the quantile function of the GOBX-G family of distributions as

Special Cases
In this section, GOBX-G family of distributions takes on some selected baseline distributions even though any baseline distribution can be considered. For our study, we consider Weibull distribution, og-ogistic distribution and Uniform distribution.

Gamma Odd Burr X-Weibull (GOBX-W) Distribution
Considering the baseline distribution of the GOBX-G family as Weibull distribution with pdf g(x; α) = αx α−1 e −x α and cdf G(x; α) = 1 − e −x α , we respectively obtain the pdf, cdf and hrf of the GOBX-W distribution as for δ, θ, α > 0. Plots of the pdf and hrf of GOBX-W distribution are given in Figure 1.
The Gamma Odd Burr X-G Family of Distributions with Applications
The Gamma Odd Burr X-G Family of Distributions with Applications

Gamma Odd Burr X-Uniform (GOBX-U) Distribution
For a uniform baseline distribution with pdf g(x; µ) = 1/µ and cdf G(x; µ) = x/µ, 0 < x < µ, we respectively obtain the pdf, cdf and hrf of the GOBX-U distribution as The Gamma Odd Burr X-G Family of Distributions with Applications for δ, θ, µ > 0. Plots of the pdf and hrf of GOBX-U distribution are given in Figure 3. Figure 3: Plots of the pdf and hrf for the GOBX-U distribution Figure 3 shows the GOBX-U distribution pdf and hrf. The pdf takes on different shapes which includes left-skew, rightskew, almost symmetric and reverse-J. Additionally, the hrf also show increasing, decreasing and bathtub shapes. Table 1 presents some values generated from the GOBX-W distribution for different values of the parameters δ, θ and α.

Other Statistical and Mathematical Properties
In this section, some structural properties of GOBX-G family of distributions are presented. These includes moments and generating functions, conditional moments, entropy, distribution of order statistics and probability weighted moments.

Moments and Generating functions
The r th ordinary moment of GOBX-G family of distributions can be derived from equation (6) as where Y p+1 is the Exp-G random variable with power parameter p + 1 and Q G (u; ξ) is the quantile function of the baseline distribution with the cdf G(x; ξ). To obtain the skewness and the kurtosis of GOBX-G family of distributions, The Gamma Odd Burr X-G Family of Distributions with Applications we use the n th central moment, say M n , given as The moment generating function of the GOBX-G family of distributions can be obtained as where M p+1 (t) is the moment generating function of the Exp-G random variable Y p+1 . Figure 4 and 5 shows 3D plots of skewness and kurtosis for the GOBX-W distribution for some selected parameter values. Figure 4 shows that, when θ is fixed, skewness increases as α increases whereas kurtosis decrease with a decrease in δ. An increase in θ leads to increase in skewness and kurtosis when α is held constant as shown in figure 5. This shows that all the three parameters play a role in the variation of skewness an kurtosis.

Incomplete Moments
The s th (s > 0) incomplete moments for the GOBX-G family of distributions follow from equation (6) as The incomplete moments are critical as they can be used to derive the mean deviations, Bonferroni, Lorenz and Zenga curves which are usually used in many fields such as demography, engineering and medicine. The mean deviation about the mean µ = E(X) and median M = Median(X) can respectively be derived as E|X − µ 1 | = 2µ 1 F(µ 1 ) − 2ϕ 1 (µ 1 ) and where µ 1 can be evaluated from equation (13), M= Q(0.5) from equation (9) and F(µ 1 ) from equation (3).

Rényi Entropy
This subsection presents the Rényi entropy as a measures of the variation of uncertainty. Rényi entropy is an extension of Shannon entropy and it is defined as in the GOBX-G density given in equation (4), then we have

Using series representation
i+1 , for y ∈ (0, 1), we have , so that f v (x; δ, θ, ξ) can be written as If we let c s = (s + 2) −1 and considering the result of a power series raised to a positive integer (Grandshteyn and Ryzhik, 1980), we get where w * = m + s + v(δ − 1) + 1. Applying the generalized binomial series representation, we get Applying the exponential series representation, we have Applying the generalized binomial series representation in G 2k+v ( Finally, the Rényi entropy for the GOBX-G family of distributions is given as The Gamma Odd Burr X-G Family of Distributions with Applications dx is the Rényi entropy for the Exp-G family with power parameter p v + 1 and

Probability Weighted Moments
This section present the probability weighted moments with most of the derivations given under the order statistics section. The (r, s) th probability weighted moments for the GOBX-G family of distributions is given as With the help of derivations from equation (14) under order statistics section, we get where y = exp − G(x;ξ) G(x;ξ) 2 and d j,k is defined in equation (15). Plugging f (x), we get The Gamma Odd Burr X-G Family of Distributions with Applications Further expansion based on equation (16) yields where b s,m is defined in equation (5) and v * = k + 2δ + m + s + r. Series expansion of where h * q+1 (x; ξ) is defined under equation (17) and Consequently, we obtain the probability weighted moments for the GOBX-G family of distributions reduces to

Maximum Likelihood Estimation
This section present the maximum likelihood technique for estimating the parameters of the GOBX-G family of distributions. Maximum likelihood is one of the most reliable estimation technique as it mostly produce accurate and The Gamma Odd Burr X-G Family of Distributions with Applications consistent estimators (Zanakis and Kyparisis, 1986). Let X 1 , X 2 , X 3 ,...,X n be a random sample of size n from the GOBX-G family of distributions and Θ = (δ, θ, ξ) T be a vector of model parameters, then the log-likelihood function can be expressed as are the diagonal elements of I −1 n (Θ) = (nI(Θ)) −1 and Z ϕ 2 is the standard normal ϕ

Simulation Study
In this section, simulation results are presented for different sample sizes of n = 100, 200, 400, 600 and 800 to examine the accuracy and consistency of the maximum likelihood estimators (MLEs) for each of the parameter of the GOBX-W distribution. The simulation was repeated N = 1000 times and the mean estimates, average bias (Abias) and the root mean square errors (RMSEs) were evaluated. For consistent MLEs, it is expected that as the sample size n increases, the mean estimates gets closer to the true parameters and, the RSMEs and Abias also decay to zero. Table  2 and 3 show the mean estimates together with their respective RSMEs and Abias. The Abias and RMSEs for the estimated parameter, say,θ, are respectively given as:  From the results in Table 2 and 3, it is clear that as the sample size increases, the mean estimates gets closer to true parameters whereas the respective RSMEs and Abias decay to zero indicating consistent MLEs.

Applications
This section presents two real data applications to show the flexibility of GOBX-W distribution compared with other existing distributions. The first data consists of n = 59 monthly the actual taxes revenue (in 1000 million Egyptian pounds) in Egypt between January 2006 and November 2010 extracted from Nassar and Nada (2011) and given as:  In addition, the Cramér-von Mises (W * ) and Andersen-Darling (A * ) as described by Chen and Balakrishnan (1995) were for a, b, α > 0, , for k, β, λ > 0, for θ, λ, δ > 0, for a, λ > 0, and for θ, α > 0 and x > 0, respectively.

Taxes Revenue Data
This subsection contain parameter estimates (standard error in parenthesis), goodness-of-fit statistics, plots of the fitted densities, empirical cdf, Kaplan-Meier and TTT, hrf plots and probability plots for the taxes revenue data.  Figure 9: Profile plots of δ, θ and α for monthly actual taxes revenue data Figure 9(a), 9(b) and 9(c) shows profile plots of the MLEs of δ, θ and α. It can be seen that the parameters attained the absolute maximum for monthly actual taxes revenue data.

Repair Time Data
This subsection contain parameter estimates (standard error in parenthesis), goodness-of-fit statistics, plots of the fitted densities, empirical cdf, Kaplan-Meier and TTT, hrf plots and probability plots for the repair time data.