# Classical and Bayesian Inference of Marshall-Olkin Extended Gompertz Makeham Model with Modeling of Physics Data

Authors: Rania A. H. Mohamed; Abdulhakim A. Al-Babtain; I. Elbatal; Ehab M. Almetwally; Hisham M. Almongy
Journal: Journal of Mathematics
 (2022)

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## Abstract

The purpose of this study is to present the Marshall- Olkin extended Gompertz MakehamMOEGM lifetime distribution, which has four parameters. As a result, we will describe some of the structural elements that are introduced for this model. The maximum likelihood approach is used to estimate the model parameters, and it is well known that likelihood estimators for unknown parameters are not always available. As a result, we examine the prior distributions, which allow for prior dependence among the components of the parameter vector, as well as the Bayesian estimators derived with respect to the squared error loss function. A Monte Carlo simulation research is carried out to examine the performance of the likelihood estimators and the Bayesian technique. Finally, we demonstrate the significance of the new model. And to conclude, we illustrate the importance of the new model by exploring some of the empirical applications of physics to show it’s flexibility and potentiality of a new model.


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## Body

## 1. Introduction

Gompertz distribution has been obtained by Gompertz [1]. It is critical in the analysis of survival periods in several areas, including marketing, gerontology, biology, and computer science. It was used to characterize human mortality, develop growth models, and create actuarial tables. The Gompertz distribution’s hazard rate function (hrf) is an increasing function used by actuaries and demographers to characterize the distribution of adult life lengths. Makeham [2] looked at the Gompertz distribution’s fit to actuarial data and found that by modifying it, he could enhance the fit. This change is now known as the Gompertz- Makeham GM distribution. The Gompertz - Makeham (GM) distribution studied by Bailey et al. [3]. The GM distribution has been frequently utilised in actuarial tables and growth models to describe human mortality.Missov and Lenart [4] discovered closed-form solutions to the life-expectancy integral in homogeneous and gamma-heterogeneous populations, as well as in the presence or absence of the Makeham factor. Chukwu and Ogunde [5] introduced Kumaraswamy Gompertz-Makeham, a five-parameter generalized version of the GM with decreasing, rising, and bathtub-shaped failure rate functions. For the GM model, Wrycza [6] developed a straightforward formulation of life table entropy.The cumulative distribution function (c.d.f.) of the Gompertz- Makeham (GM) distribution is given by(1)GGMx,θ,α,λ=1−e−θx−α/λeλx−1,x>0,where λ>0 is a scale parameter, θ>0 and α>0 are shape parameters.The corresponding probability density function (p.d.f) and hrf are given by(2)gGMx,θ,α,λ=αeλx+θe−θx−α/λeλx−1,hGMx,θ,α,λ=gGMx,θ,α,λG¯GMx,θ,α,λ=αeλx+θ,respectively.There has lately been a resurgence of interest in developing innovative generators for univariate continuous distributions by introducing one or more additional shape factors into the baseline model. This parameter induction has been demonstrated to be useful in analyzing tail characteristics and increasing the goodness-of-fit of the recommended generator family. These asymmetric distributions were formed by adding new parameters to a baseline c.d.f., resulting in a new family of more analytically flexible asymmetric distributions. In the statistical literature, several classes have been proposed for constructing new distributions by adding one or more parameters. The beta-G by Eugene et al. [7], Kumaraswamy-G by Cordeiro and de Castro [8], new extended cosine-G distributions by Muhammad et al. [9], new truncated muth generated family by Almarashi et al. [10], odd Perks-G class by Elbatal et al. [11], and the Zografos-Balakrishnan-G family by Nadarajah et al. [12] are just a few examples of well-known generators.Marshall and Olkin [13] suggested a general approach for adding a new positive shape parameter to a baseline distribution, resulting in the Marshall-Olkin family of distributions (abbreviated as “MO” for short). The baseline distribution is included in this family as a fundamental instance, and some distributions have more flexibility for representing diverse types of data. The proportional odds family with tilt parameter are other names for the MO family of distributions (Marshall and Olkin [14]). The Marshall and Olkin family’s c.d.f. is defined as:(3)Fx,γ=Gx1−γ¯G¯x,x>0,γ>0.The survival functionF¯x,γ is given by(4)F¯x,γ=γG¯x1−γ¯G¯x,where γ¯=1−γ, for γ=1, we get the baseline distribution, i.e., F¯x=G¯x, where the shape parameter γ is called tilt parameter, since the hazard rate function hx;γ of the transformed distribution is shifted below when γ≥1 or shifted below when 0<γ≤1 from the baseline hazard rate function hGx. In fact, hx;γ≤hGx when γ≥1 and hx;γ≥hGx when 0<γ≤1. The corresponding p.d.f becomes(5)fx,γ=γgx1−γ¯G¯x2,the hrf is given by(6)hx,γ=fx,γF¯x,γ=γgxG¯x1−γ¯G¯x.In recent years, several authors have used this method to extend well-known distributions. A few examples include Ghitany et al. [15] presented censored scheme of MO extended Weibull distribution, Jayakumar and Mathew [16] introduced on a generalization to MO with application of Burr type XII distribution, Pérez-Casany and Casellas [17] presented MO extended Zipf Distribution, Krishna et al. [18] proposed the MO Fréchet distribution, Gui [19] introduced the MO power log - normal distribution and and its applications to survival data, Idika et al. [20] introduced the MO generalized Erlange - truncated exponential distribution, MirMostafaee et al. [21] represented the MO extended generalized Rayleigh distribution, among others. The aim of this paper is to propose a new class of lifetime distributions called “The MO extended Gompertz-Makeham” distribution, as referred to as MOEGM.In this paper, the Marshall- Olkin extended Gompertz MakehamMOEGM lifetime distribution has been presented, which has four parameters. As a result, we will describe some of the structural properties that are introduced for this model. The maximum likelihood approach is used to estimate the model parameters, and it is well known that likelihood estimators for unknown parameters are not always available. As a result, we examine the prior distributions, which allow for prior dependence among the components of the parameter vector, as well as the Bayesian estimators derived with respect to the squared error loss function. A Monte Carlo simulation and two real data sets are carried out to examine the performance of the model and likelihood estimators and the Bayesian technique.The rest of the paper is organized as follows: The Marshall-Olkin extended Gompertz Makeham distribution and its technique are defined in Section2. Section 3 introduces and investigates numerous structural characteristics properties of the MOEGM distribution. Section 4 shows the likelihood estimates for the unknown parameters. Section 5 shows the Bayesian estimates of the unknown parameters. Simulation results are carried out in Section 6. Section 7 depicts two real-world data applications. Finally, we demonstrate the significance of this study’s closing remarks.


## 2. TheMOEGM Model

In this section, we introduce the four parameter Marshall- Olkin extended Gompertz-MakehamMOEGM distribution. Using equations (1), (3) and (4) shown in the previous section, the c.d.f. and survival function can be written as follows,(7)Fx;θ,α,λ,γ=Gx1−γ¯G¯x=1−e−θx−α/λeλx−11−γ¯e−θx−α/λeλx−1,F¯x,θ,α,λ,γ=G¯x1−γ¯G¯x=e−θx−α/λeλx−11−γ¯e−θx−α/λeλx−1,respectevely. The corresponding p.d.f given by(8)fx;θ,α,λ,γ=γαeλx+θe−θx−α/λeλx−11−γ¯e−θx−α/λeλx−12.Henceforth, LetX∼MOEGMφ, having p.d.f. (8) where φ=θ,α,λ,γ. Figure 1 display some plots of the p.d.f. of MOEGM model for some different parameter values.Figure 1
The p.d.f. plot for theMOEGM model.The failure (hazard) rate function in event time analysis quantifies the current likelihood of failure for the population that has not yet failed. The hrf is essential when dealing with lifetime data in reliability analysis, survival analysis, and demography, as well as when building and creating models. The hrf for the Marshall-Olkin extended Gompertz-Makeham distribution is as follows in Figure2. Figure 2 display some plots of the hrf of MOEGM model for some different parameter values.(9)hx,φ=fx;φF¯x;φ=γαeλx+θ1−γ¯e−θx−α/λeλx−1.Figure 2
The hrf plot for theMOEGM model.


### 2.1. Expansion of p.d.f

In this subsection, we present the expansion of theMOEGM density function in terms of an infinite linear combination of Gompertz-Makeham distribution. using the power series expansion(10)1−z−n=∑i=0∞Γn+iΓni!zi,n>0,z<1.We get(11)1−γ¯e−θx−α/λeλx−1−2=∑i=0∞i+1γ¯ie−iθx+α/λeλx−1,substituting equation (11) into equation (8), we get(12)fx;φ=γ∑i=0∞i+1γ¯iαeλx+θe−i+1θx+α/λeλx−1.Using the series expansion ofe−i+1eλx as follows(13)e−i+1α/λeλx=∑j=0∞−1ji+1jα/λjj!eλjx,thus after some algebra (12) can be written as(14)fx;φ=∑k=0∞δkαj+1k+θjkxke−i+1θx,where(15)δk=∑i,j=0∞γi+1j+1−1j+kγ¯iei+1α/λα/λjλkj!k!.


## 2.1. Expansion of p.d.f

In this subsection, we present the expansion of theMOEGM density function in terms of an infinite linear combination of Gompertz-Makeham distribution. using the power series expansion(10)1−z−n=∑i=0∞Γn+iΓni!zi,n>0,z<1.We get(11)1−γ¯e−θx−α/λeλx−1−2=∑i=0∞i+1γ¯ie−iθx+α/λeλx−1,substituting equation (11) into equation (8), we get(12)fx;φ=γ∑i=0∞i+1γ¯iαeλx+θe−i+1θx+α/λeλx−1.Using the series expansion ofe−i+1eλx as follows(13)e−i+1α/λeλx=∑j=0∞−1ji+1jα/λjj!eλjx,thus after some algebra (12) can be written as(14)fx;φ=∑k=0∞δkαj+1k+θjkxke−i+1θx,where(15)δk=∑i,j=0∞γi+1j+1−1j+kγ¯iei+1α/λα/λjλkj!k!.


## 3. Statistical Features

### 3.1. Quantile Function

For a random variableX has c.d.f. of Marshall- Olkin power generalized Weibull distribution, the quantile function Qp is given by the relation(16)θxp+αλeλxp−1+log1−p1−pγ¯=0,p∈0,1.By equation (16), in addition to using the qf to obtain the Bowley’s skewness and the Moors’ kurtosis, is highly useful for generating MOEGM random variate and can be simply applied. Bowley’s skewness is based on quartiles, as described by Kenney and Keeping [22], it’s given by(17)BS=Q3/4−2Q2/4+Q1/4Q3/4−Q1/4,and the Moor’s kurtosis, see Moors [23], is given by(18)Mk=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8,where Q⋅ is the quantile function given by equation (16).


### 3.2. Moments

Therth moment of the MOEGM distribution is discussed in this subsection. In any statistical analysis, especially in applications, moments are crucial and important. It can be used to investigate a distribution’s most essential properties and qualities (e.g., tendency, dispersion, skewness and kurtosis).


### 3.3. Theorem Quantile Function

IfX has X∼MOEGMφ, where φ=θ,α,λ,γ then the rth moment of X is given by(19)μr′x=∑k=0∞δkαj+1k+θjkΓr+k+1i+1θr+k+1.Proof.
LetX be a random variable with the distribution MOEGM. The well-known formula can be used to calculate the rth ordinary moment.(20)μr′x=∫0∞xrfx,φdx=∑k=0∞δkαj+1k+θjk∫0∞xr+ke−i+1θxdx,setting y=i+1θx, after some algebra, the rth ordinary moment can be written as(21)μr′x=∑k=0∞δkαj+1k+θjkΓr+k+1i+1θr+k+1,where Γn=∫0∞xn−1e−xdx denotes the gamma function.


### 3.4. Moment Generating Function

Moment generating functions are helpful for a variety of reasons, one of which being their usage in sums of random variables analysis. When compared to working directly with the probability function or c.d.f. of a random variable, it provides the foundation for an alternative approach to analytic solutions.Theorem 1.
IfX has the MOEGMθ,α,λ,γ, then the the moment generating function (mgf) of X is given as follows(22)MXt=∑k=0∞δkαj+1k+θjkΓk+1θi+1−tk+1.Proof.
We begin with the well-known simplification of the moment generating function, which is as follows:(23)MXt=EetX=∫0∞etxfxdx=∑k=0∞δkαj+1k+θjk∫0∞xke−i+1θ−txdx=∑k=0∞δkαj+1k+θjkΓk+1θi+1−tk+1,which completes the proof.


### 3.5. Conditional Moments

Thesth lower incomplete moment of MOEGM distribution is(24)ηst=∫0txsfxdx=∑k=0∞δkαj+1k+θjk∫0txs+ke−i+1θxdx=∑k=0∞δkαj+1k+θjkγs+k+1,θi+1tθi+1s+k+1,where γs,t=∫0txs−1e−xdx is the lower incomplete gamma function. The first incomplete moment of X, denoted by, η1t, is computed using equation (24) by setting s = 1 as(25)η1t=∑k=0∞δkαj+1k+θjkγk+2,θi+1tθi+1k+2.Similarly, thesth upper incomplete moment of MOEGM distribution is(26)ξst=∫t∞xsfxdx=∑k=0∞δkαj+1k+θjk∫t∞xs+ke−i+1θxdx=∑k=0∞δkαj+1k+θjkΓs+k+1,θi+1tθi+1s+k+1,where Γs,t=∫t∞xs−1e−xdx is the upper incomplete gamma function.The mean residual lifespanMRL has a diverse set of uses and applications see Lai and Xie [24]. The expected extended life length for a unit alive at age t is represented by the MRL (or life expectancy at age t). The MRL is given by(27)μt=EX|X>t=ξ1tF¯t−t,where ξ1t is the first incomplete moment of X and by setting s=1 in equation (26), we get(28)μt=1F¯t∑k=0∞δkαj+1k+θjkΓk+2,θi+1tθi+1k+2−t.In addition, the mean inactivity timeMIT shows the amount of time that has passed after an item has failed, assuming that the failure happened in 0;t. For t>0, the MIT of X is defined by(29)τt=EX|X<t=t−η1tFt=t−1Ft∑k=0∞δkαj+1k+θjkγk+2,θi+1tθi+1k+2.


## 3.1. Quantile Function

For a random variableX has c.d.f. of Marshall- Olkin power generalized Weibull distribution, the quantile function Qp is given by the relation(16)θxp+αλeλxp−1+log1−p1−pγ¯=0,p∈0,1.By equation (16), in addition to using the qf to obtain the Bowley’s skewness and the Moors’ kurtosis, is highly useful for generating MOEGM random variate and can be simply applied. Bowley’s skewness is based on quartiles, as described by Kenney and Keeping [22], it’s given by(17)BS=Q3/4−2Q2/4+Q1/4Q3/4−Q1/4,and the Moor’s kurtosis, see Moors [23], is given by(18)Mk=Q7/8−Q5/8+Q3/8−Q1/8Q6/8−Q2/8,where Q⋅ is the quantile function given by equation (16).


## 3.2. Moments

Therth moment of the MOEGM distribution is discussed in this subsection. In any statistical analysis, especially in applications, moments are crucial and important. It can be used to investigate a distribution’s most essential properties and qualities (e.g., tendency, dispersion, skewness and kurtosis).


## 3.3. Theorem Quantile Function

IfX has X∼MOEGMφ, where φ=θ,α,λ,γ then the rth moment of X is given by(19)μr′x=∑k=0∞δkαj+1k+θjkΓr+k+1i+1θr+k+1.Proof.
LetX be a random variable with the distribution MOEGM. The well-known formula can be used to calculate the rth ordinary moment.(20)μr′x=∫0∞xrfx,φdx=∑k=0∞δkαj+1k+θjk∫0∞xr+ke−i+1θxdx,setting y=i+1θx, after some algebra, the rth ordinary moment can be written as(21)μr′x=∑k=0∞δkαj+1k+θjkΓr+k+1i+1θr+k+1,where Γn=∫0∞xn−1e−xdx denotes the gamma function.


## 3.4. Moment Generating Function

Moment generating functions are helpful for a variety of reasons, one of which being their usage in sums of random variables analysis. When compared to working directly with the probability function or c.d.f. of a random variable, it provides the foundation for an alternative approach to analytic solutions.Theorem 1.
IfX has the MOEGMθ,α,λ,γ, then the the moment generating function (mgf) of X is given as follows(22)MXt=∑k=0∞δkαj+1k+θjkΓk+1θi+1−tk+1.Proof.
We begin with the well-known simplification of the moment generating function, which is as follows:(23)MXt=EetX=∫0∞etxfxdx=∑k=0∞δkαj+1k+θjk∫0∞xke−i+1θ−txdx=∑k=0∞δkαj+1k+θjkΓk+1θi+1−tk+1,which completes the proof.


## 3.5. Conditional Moments

Thesth lower incomplete moment of MOEGM distribution is(24)ηst=∫0txsfxdx=∑k=0∞δkαj+1k+θjk∫0txs+ke−i+1θxdx=∑k=0∞δkαj+1k+θjkγs+k+1,θi+1tθi+1s+k+1,where γs,t=∫0txs−1e−xdx is the lower incomplete gamma function. The first incomplete moment of X, denoted by, η1t, is computed using equation (24) by setting s = 1 as(25)η1t=∑k=0∞δkαj+1k+θjkγk+2,θi+1tθi+1k+2.Similarly, thesth upper incomplete moment of MOEGM distribution is(26)ξst=∫t∞xsfxdx=∑k=0∞δkαj+1k+θjk∫t∞xs+ke−i+1θxdx=∑k=0∞δkαj+1k+θjkΓs+k+1,θi+1tθi+1s+k+1,where Γs,t=∫t∞xs−1e−xdx is the upper incomplete gamma function.The mean residual lifespanMRL has a diverse set of uses and applications see Lai and Xie [24]. The expected extended life length for a unit alive at age t is represented by the MRL (or life expectancy at age t). The MRL is given by(27)μt=EX|X>t=ξ1tF¯t−t,where ξ1t is the first incomplete moment of X and by setting s=1 in equation (26), we get(28)μt=1F¯t∑k=0∞δkαj+1k+θjkΓk+2,θi+1tθi+1k+2−t.In addition, the mean inactivity timeMIT shows the amount of time that has passed after an item has failed, assuming that the failure happened in 0;t. For t>0, the MIT of X is defined by(29)τt=EX|X<t=t−η1tFt=t−1Ft∑k=0∞δkαj+1k+θjkγk+2,θi+1tθi+1k+2.


## 4. Estimation and Inference

Only full samples are used to calculate the maximum likelihood estimates (MLEs) of the parameters of theMOEGM distribution in this section. Let X1,…,Xn be a random sample of size n from MOEGMφ where φ=θ,α,λ,γ. Let φ=θ,α,λ,γT be the parameter vector. The log-likelihood function for the vector of parameters φ=θ,α,λ,γ can be written as(30)logLφ=nlogγ+∑i=1nlogαeλxi+θ−θ∑i=1nxi−αλ∑i=1nlogeλxi−1−2∑i=1nlog1−γ¯e−θxi−α/λeλxi−1.The following is the associated score function:(31)Unφ=∂Lφ∂θ,∂Lφ∂α,∂Lφ∂λ,∂Lφ∂γT.Either directly or by solving the nonlinear likelihood equations derived by differentiating equation (30), the log-likelihood can be maximized. The score vector’s components of likelihood are as follows:(32)∂logLφ∂θ=∑i=1n1αeλxi+θ−∑i=1nxi−2∑i=1nγ¯xie−θxi−α/λeλxi−11−γ¯e−θxi−α/λeλxi−1,∂logLφ∂α=∑i=1neλxiαeλxi+θ−1λ∑i=1nlogeλxi−1−2λ∑i=1nγ¯e−θxi−α/λeλxi−1eλxi−11−γ¯e−θxi−α/λeλxi−1,∂logLφ∂λ=∑i=1nαxieλxiαeλxi+θ−α∑i=1nxieλxiλ−eλxi−1λ2+2αλ2∑i=1nγ¯e−θxi−α/λeλxi−1λxieλxi−eλxi+11−γ¯e−θxi−α/λeλxi−1,∂logLφ∂γ=nγ−2∑i=1ne−θxi−α/λeλxi−11−γ¯e−θxi−α/λeλxi−1.The maximum likelihood estimation (MLE) ofφ, say φ^, is obtained by solving the nonlinear system Unφ=0.


## 5. Bayesian Estimation

The Bayesian technique deals with the parameters because random and parameter uncertainties are represented by a previous joint distribution that was formed before the failure data was collected. The flexibility of the Bayesian technique to incorporate past knowledge into research makes it particularly useful in the study of reliability, as one of the major challenges with reliability analysis is a lack of data. Prior gamma distributions are used in theθ,λ,α and γ parameters of the MOEGM distribution, where θ,λ,α, and γ are non-negative values. As separate joint prior density functions, the θ,λ,α, and γ parameters as follows:(33)∏θ,λ,α,γ∝θq1−1λq2−1αq3−1γq4−1e−w1θ+w2λ+w3α+w4γ.The likelihood function of theMOEGM distribution and joint prior density (30) are used to produce the joint posterior density function of θ,λ,α, and γ.(34)πθ,λ,α,γ|x¯=Lx¯|θ,λ,α,γ∏θ,λ,α,γ∫θ∫λ∫α∫γLx¯|θ,λ,α,γ∏θ,λ,α,γdθdλdαdγ,∝Lx¯|θ,λ,α,γ∏θ,λ,α,γ,∝θq1−1λq2−1αq3−1γn+q4−1e−θw1+∑i=1nxi−αw3+∑i=1neλxi−1/λ×∏i=1ne−w4γαe−λw2−xi+θe−w2λ1−γ¯e−θxi−α/λeλxi−12.The majority of Bayesian inference algorithms are based on symmetric loss functions. A prominent symmetric loss function is the squared-error loss function (SELF). The Bayesian estimators ofθ,λ,α, and γ, say θ˜B,λ˜B,α˜B,γ˜B based on SELF.(35)θ˜=Eθ|α,λ,γ,x¯,λ˜=Eλ|α,θ,γ,x¯,α˜=Eα|θ,λ,γ,x¯,γ˜=Eγ|θ,λ,α,x¯.It should be noted that the integrals supplied by equation (35) cannot be deduced clearly. As a result, we use Markov-Chain-Monte-Carlo (MCMC) to approximate the value of expectations in equation (35).An observation was made that the integrals are given by equation (35) are not possible to derive explicitly. As a result, we employ the MCMC technique to approximate the value of integrals in equation (35). Many of studies used MCMC technique such as Al-Babtain et al. [25], Tolba et al. [26, 27], and Bantan et al. [28].In Gibbs samplers, more general Metropolis algorithms are important subclasses of MCMC algorithms. Two of the most prevalent MCMC methodologies are the Metropolis-Hastings (MH) and Gibbs sampling methods. The MH technique, like acceptance-rejection sampling, assumes that each algorithm iteration can yield a candidate value from a proposal distribution. We apply the MH in the Gibbs sampling phases to get random samples of conditional posterior densities from theMOEGM distribution:(36)πθ|λ,α,γ,x¯∝θn+q1−1e−θw1+∑i=1nxi∏i=1n11−γ¯e−θxi−α/λeλxi−12,πλ|θ,α,γ,x¯∝λq2−1e−α∑i=1neλxi−1/λ∏i=1ne−λw2−xi+θe−w2λ1−γ¯e−θxi−α/λeλxi−12,πα|θ,λ,γ,x¯∝αq3−1e−αw3∏i=1nα1−γ¯e−θxi−α/λeλxi−12,πγ|θ,λ,α,x¯∝γn+q4−1∏i=1ne−w4γ1−γ¯e−θxi−α/λeλxi−12.


## 6. Simulation

The Monte-Carlo simulation approach is used in this section to compare the likelihood estimation method with the Bayesian estimation method. TheR language is used to estimate MOEGM distribution parameters using MLE and a Bayesian estimation approach based on MCMC under SELF. Monte-Carlo experiments are carried out using 10000 randomly generated MOEGM distribution samples, where x represents the MOEGM lifetime for various parameter actual values and sample sizes n: 30,70,150,and200. The best estimator approaches could be described as minimizing estimator bias (A1) and mean squared error (A2). The MOEGM distribution’s true parameters have been determined.Tables1–3 describe the simulation results of the approaches presented in this paper for point estimate. In order to do the essential comparison between various point estimating methods, we examine the A1 and A2 values. As a result, the following conclusions were drawn:(1)
For parameters of theMOEGM distribution, the A1 and A2 decrease as sample size n grows.(2)
The best estimating method is Bayesian estimation.(3)
TheA1 and A2 for all parameters diminish as γ increases.(4)
TheA1 and A2 for all parameters increase as θ increases.Table 1
A1 and A2 of MOEGM parameters by MLE, and Bayesian when. α=0.5,λ=0.5.
γ = 0.75
γ = 3
α
=
0.5
,
λ
=
0.5
MLE
Bayesian
MLE
Bayesian
θ
n
A1
A2
A1
A2
A1
A2
A1
A2
0.75
30
θ
0.0663
0.2090
−0.0105
0.0807
0.1024
0.1833
−0.0064
0.0839
α
0.2976
2.6962
−0.0118
0.0642
−0.0389
0.3818
0.0092
0.0688
λ
0.5169
2.1535
−0.0483
0.0773
0.4304
1.0915
−0.0201
0.0679
γ
0.6221
5.1127
−0.0147
0.0593
0.2325
3.3815
−0.0333
0.1281
70
θ
0.0361
0.1570
−0.0009
0.0252
0.1016
0.1289
0.0094
0.0244
α
0.0892
1.0049
−0.0178
0.0238
−0.0932
0.1873
0.0020
0.0174
λ
0.3333
1.1596
−0.0219
0.0284
0.3409
0.4543
−0.0251
0.0271
γ
0.2307
1.4412
−0.0108
0.0224
−0.0001
1.3741
−0.0164
0.0293
150
θ
0.0413
0.1089
0.0009
0.0204
0.0643
0.0681
−0.0001
0.0187
α
−0.0081
0.3685
−0.0046
0.0162
−0.0659
0.1047
0.0050
0.0119
λ
0.2005
0.5740
−0.0189
0.0251
0.2062
0.2413
−0.0174
0.0172
γ
0.0851
0.5711
−0.0052
0.0172
0.0176
0.6182
0.0087
0.0219
200
θ
0.0605
0.0764
0.0019
0.0056
0.0504
0.0349
−0.0053
0.0051
α
−0.0431
0.0853
−0.0059
0.0048
−0.0346
0.0603
0.0026
0.0045
λ
0.1002
0.2251
−0.0083
0.0062
0.1023
0.1092
−0.0004
0.0045
γ
0.0373
0.1438
−0.0064
0.0049
0.0306
0.4561
−0.0045
0.0064
3
30
θ
−0.6898
1.7822
−0.0026
0.1054
−0.4916
1.0311
−0.0101
0.1357
α
0.9339
10.4192
−0.0348
0.0685
0.5875
5.2078
−0.0263
0.0934
λ
1.5259
10.5863
−0.0581
0.1057
0.9608
5.9980
−0.0252
0.0891
γ
0.3771
2.7854
0.0451
0.0909
0.8456
9.9552
−0.0161
0.1125
70
θ
−0.6936
1.3025
−0.0004
0.0283
−0.4240
0.7049
−0.0079
0.0323
α
0.5995
5.5597
−0.0336
0.0319
0.7208
4.3036
−0.0227
0.0276
λ
0.8973
4.5855
−0.0218
0.0362
0.4344
3.7110
−0.0188
0.0314
γ
0.0863
0.9616
−0.0041
0.0221
0.7759
9.5984
0.0030
0.0353
150
θ
−0.4045
0.7562
−0.0024
0.0246
−0.2864
0.4528
−0.0088
0.0243
α
0.5395
3.1483
−0.0043
0.0252
0.5922
2.7309
−0.0113
0.0204
λ
0.3132
2.7684
−0.0180
0.0268
0.1767
2.6308
−0.0174
0.0270
γ
0.1129
0.6149
0.0007
0.0130
0.5947
5.4817
0.0037
0.0255
200
θ
−0.3160
0.4131
0.0007
0.0060
−0.1378
0.2110
0.0019
0.0070
α
0.2804
0.8734
−0.0102
0.0067
0.2864
0.9394
−0.0035
0.0055
λ
0.1636
1.3895
−0.0075
0.0081
0.0428
1.4592
−0.0057
0.0072
γ
0.0105
0.1054
0.0013
0.0043
0.2357
2.0228
−0.0033
0.0065Table 2
A1, and A2 of MOEGM parameters by MLE, and Bayesian when.α=2,λ=0.75.
γ = 0.75
γ = 3
α
=
2
,
θ
=
0.75
MLE
Bayesian
MLE
Bayesian
λ
n
A1
A2
A1
A2
A1
A2
A1
A2
0.5
30
θ
0.2234
1.3192
−0.0571
0.1101
0.1057
0.5171
−0.0292
0.0957
α
0.2155
8.8678
0.0025
0.1119
−0.2657
2.6640
0.0056
0.1101
λ
0.7288
6.6163
−0.0177
0.0883
0.6797
2.8948
−0.0474
0.0687
γ
0.5717
5.5573
0.0139
0.0535
0.6411
9.8986
−0.0285
0.1085
70
θ
0.0565
0.9048
−0.0155
0.0298
0.0954
0.3111
−0.0119
0.0304
α
0.0168
3.6365
−0.0194
0.0301
−0.1759
1.2053
−0.0051
0.0262
λ
0.2718
2.7239
−0.0203
0.0316
0.3094
1.1250
−0.0149
0.0300
γ
0.2262
1.9272
0.0014
0.0197
0.3733
7.0686
−0.0003
0.0335
150
θ
0.0826
0.6743
−0.0187
0.0269
0.0326
0.1921
−0.0013
0.0216
α
0.1547
3.0759
−0.0044
0.0227
−0.2055
0.4865
−0.0038
0.0205
λ
0.0107
1.8665
−0.0170
0.0246
0.2044
0.4764
−0.0012
0.0200
γ
0.2081
1.3476
−0.00001
0.0155
−0.0034
2.7353
0.0001
0.0258
200
θ
0.0121
0.3445
0.00004
0.0064
0.0015
0.0763
−0.0033
0.0063
α
−0.0127
0.9029
−0.0012
0.0058
−0.1218
0.2391
0.0017
0.0059
λ
0.0517
0.6778
−0.0063
0.0066
0.1161
0.1799
−0.0056
0.0053
γ
0.0401
0.3068
0.0013
0.0043
−0.0460
1.2654
−0.0002
0.0059
3
30
θ
0.5940
2.3168
−0.0579
0.1390
0.1581
1.38285
−0.0576
0.11203
α
0.0865
5.9463
−0.0084
0.0992
−0.2814
2.56117
0.0197
0.10938
λ
0.9393
9.0583
−0.0141
0.1159
1.1403
5.1957
−0.0045
0.1068
γ
0.5021
3.3756
0.0013
0.0544
0.2552
9.34188
−0.0265
0.12716
70
θ
0.4157
1.2562
−0.0165
0.0327
0.1110
0.45985
−0.0051
0.03112
α
0.0020
3.1157
−0.0081
0.0297
−0.1964
1.6083
−0.0062
0.0274
λ
0.4155
3.8968
−0.0107
0.0295
0.6467
2.39636
−0.0168
0.02602
γ
0.2856
1.5833
−0.0040
0.0207
0.1558
4.69425
−0.0031
0.03366
150
θ
0.2229
0.6404
−0.0268
0.0249
0.1088
0.1939
−0.0139
0.0266
α
−0.1318
1.2423
0.0052
0.0212
−0.1068
0.99542
0.0015
0.02033
λ
0.3486
2.0025
−0.0055
0.0249
0.3275
1.06376
−0.0004
0.02312
γ
0.1034
0.6022
0.0004
0.0142
0.1821
2.7484
−0.0116
0.0256
200
θ
0.0952
0.2498
−0.0003
0.0058
0.0622
0.06397
−0.0057
0.00671
α
−0.1570
0.5557
−0.0010
0.0061
−0.0534
0.58078
−0.0028
0.00549
λ
0.2533
0.7987
0.0012
0.0061
0.1733
0.5270
−0.0043
0.0058
γ
0.0112
0.1641
0.0020
0.0042
0.1292
1.52845
−0.0012
0.00647Table 3
A1 and A2 of MOEGM parameters by MLE, and Bayesian when.α=1.5,λ=2.
γ = 0.75
γ = 3
θ
=
1.5
,
λ
=
2
MLE
Bayesian
MLE
Bayesian
α
n
A1
A2
A1
A2
A1
A2
A1
A2
2
30
θ
0.3758
2.0520
−0.0504
0.1184
0.1751
1.3836
−0.0540
0.1333
α
0.1679
7.5009
−0.0406
0.1219
−0.0944
3.4698
−0.0152
0.1037
λ
1.2045
10.8415
−0.0046
0.1205
1.0588
6.0416
0.0114
0.1023
γ
0.4812
3.8034
0.0184
0.0523
0.8747
9.6510
−0.0330
0.1342
70
θ
0.1450
1.1987
−0.0010
0.0338
0.2552
0.7051
−0.0007
0.0292
α
−0.1459
2.9704
−0.0025
0.0295
−0.1261
1.7495
−0.0012
0.0275
λ
0.7292
5.2783
−0.0206
0.0307
0.5441
2.7962
−0.0113
0.0273
γ
0.0962
0.7837
0.0040
0.0190
0.5367
7.7801
−0.0064
0.0333
150
θ
0.1380
0.7847
0.0020
0.0247
0.1120
0.2308
−0.0063
0.0263
α
−0.0373
1.3426
−0.0046
0.0252
−0.1260
0.9165
−0.0007
0.0208
λ
0.2790
2.4414
−0.0041
0.0288
0.3392
1.2381
−0.0091
0.0231
γ
0.0879
0.4284
0.0054
0.0142
0.2194
3.2863
−0.0066
0.0243
200
θ
0.0384
0.3915
−0.0017
0.0065
0.0734
0.1321
−0.0037
0.0061
α
−0.1123
0.4354
0.0009
0.0069
−0.0612
0.5020
0.0042
0.0064
λ
0.2016
0.9034
−0.0012
0.0064
0.1647
0.5829
−0.0015
0.0059
γ
0.0069
0.1181
−0.0038
0.0046
0.1241
1.5795
0.0004
0.0066
4
30
θ
0.2236
3.2384
−0.0636
0.1240
0.2958
2.5922
−0.0100
0.1290
α
−0.1682
7.2254
−0.0192
0.1180
−0.6578
7.8654
−0.0266
0.1177
λ
1.1611
9.1960
−0.0561
0.1317
1.5862
9.9355
0.0020
0.1226
γ
0.2173
1.4663
0.0239
0.0529
0.9513
9.4847
−0.0503
0.1214
70
θ
0.2432
2.8862
−0.0067
0.0291
0.2506
1.9412
−0.0095
0.0343
α
0.0337
5.1864
−0.0036
0.0326
−0.5213
4.9119
−0.0199
0.0336
λ
0.3411
6.1987
−0.0065
0.0278
0.8382
5.8843
−0.0292
0.0346
γ
0.1539
0.7071
0.0074
0.0197
0.4678
10.9169
−0.0150
0.0357
150
θ
0.0727
1.3884
−0.0028
0.0246
0.0395
0.9230
−0.0098
0.0262
α
−0.1128
1.7539
−0.0143
0.0257
−0.3259
2.2341
0.0030
0.0261
λ
0.2263
2.2326
−0.0131
0.0249
0.4201
2.2449
−0.0039
0.0222
γ
0.0540
0.2351
0.0067
0.0117
0.0980
3.8122
−0.0031
0.0239
200
θ
0.0722
0.9579
0.0036
0.0074
0.0215
0.6763
0.0000
0.0060
α
−0.0290
1.1019
−0.0035
0.0062
−0.2542
1.3980
−0.0041
0.0062
λ
0.0586
1.4141
0.0012
0.0065
0.2818
1.1992
−0.0040
0.0073
γ
0.0331
0.1423
0.0021
0.0043
0.0397
2.1774
0.0021
0.0069


## 7. Applications of Physics

In this section, two real-world data applications are used to demonstrate the significance of theMOEGM distribution. We employ the Akaike information criterion measures (AICM), Bayesian information criterion measures (BICM), Consistent Akaike information Criterion (AICCM), Kolmogorov-Smirnov statistics (KSS), and the PVKSS test to compare the models. Smaller values of these statistical metrics equate to a better fit to the data set. The maximum likelihood approach is used to estimate the parameters of each distribution, while the Bayesian estimation method is used to estimate the parameters of the MOEGM distribution.


### 7.1. First Real Data of Flood Peaks

In this subsection, the first application of real data set is employed to illustrate the importance of theMOEGM distribution. This data set represents 72 excrescences of flood peaks for the years 1958–1984 (rounded to one decimal place) of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. The first data set is: “1.7, 2.5, 27.4, 1.0, 27.1, 2.2, 22.9, 1.7, 0.1, 1.1, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.9, 13.0, 12.0, 9.3, 1.4, 18.7, 8.5, 25.5, 11.6, 21.5, 27.6, 36.4, 2.7, 14.1, 22.1, 1.1, 2.5, 14.4, 1.7, 37.6, 0.6, 2.2, 39.0, 0.3, 15.0, 11.0, 7.3, 0.6, 9.0, 1.7, 7.0, 20.1, 0.4, 2.8, 14.1, 9.9, 10.4, 10.7, 30.0, 3.6, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4, 11.9, 64.0, 1.5, 20.2, 16.8, 5.3, 9.7, 27.5, 2.5 and 7.0.” The fit of the proposed model are compared with the transmuted Gompertz-Makeham (TGM) (Abd El-Bar [29]), beta generalized Gompertz (BGG) (Benkhelifa [30]), kumaraswamy gompertz makeham (KGM) (Chukwu and Ogunde [5]), Gompertz Lomax (GL) (Oguntunde et al. [31]), exponentiated generalized Weibull-Gompertz (EGWG) (El-Bassiouny et al. [32]), generalized Gompertz (GG) (El-Gohary et al. [33]) and Gompertz models.Table4 presents the MLEs with standard error (SE) of the model parameters for the first data set. The values of AICM, BICM, AICCM, HQICM, KSS and the PVKS are presented for the MOEGM model and the other models.Table 4
MLEs of the models parameters with SE and different measures for fitting: flood peaks data.
MLEs
SE
KSS
PVKS
CVMS
ADS
AICM
AICCM
BICM
HQICM
MOEGM
θ
0.0141
0.0569
0.0985
0.4873
0.1014
0.5756
506.9077
507.5047
516.0144
510.5331
α
0.0185
0.0379
λ
0.0319
0.0382
γ
0.2966
0.3969
GL
θ
1.7323
3.7299
0.0999
0.4691
0.1193
0.6709
507.6678
508.2648
516.7744
511.2931
α
0.4989
3.6786
λ
1.2268
5.6233
γ
0.1877
0.6431
Gompertz
λ
0.0013
0.0081
0.1463
0.0918
0.1120
0.6459
508.2565
508.429
518.8094
512.8534
α
0.0852
0.0148
EGWG
θ
1.6924
19.7580
0.1026
0.4351
0.1027
0.6039
509.3512
510.2603
520.7345
513.8829
λ
0.4178
4.5617
α
0.1118
4.4805
γ
0.5656
3.0847
β
0.3331
0.4674
BGG
λ
0.0123
0.0191
0.0998
0.4799
0.4754
0.5876
508.5600
509.4691
519.9433
513.0917
α
0.0331
0.0851
γ
0.5632
1.6298
θ
1.4283
4.6792
β
1.6742
3.1784
KGM
λ
0.0229
0.0442
0.1034
0.4251
0.1102
0.5928
508.7930
509.7020
520.1763
513.3247
α
0.0160
0.0508
γ
0.0399
0.0788
a
0.7910
0.1410
b
1.1351
0.9661
TGM
λ
0.0036
0.0596
0.1290
0.1821
0.1112
0.6376
508.4915
509.0885
517.5982
512.1169
α
0.0655
0.0499
θ
0.0053
0.0060
γ
0.2544
0.4056From Table4, we conclude that the MOEGM model gives the best fit, where the values of AICM, BICM, AICCM, HQICM, and KSS are smaller and the PVKS is higher for the MOEGM model when compared with those values of the other models. Figures 3(a), 3(b) illustrate the p.d.f., empirical c.d.f.s and probability plots, respectively, of the comparative models to show the over fitting of the MOEGM distribution. Figures 3(a)–3(c) illustrate estimated p.d.f. with histogram, estimated c.d.f. with empirical c.d.f., and Q-Q plot of the MOEGM distribution, respectively. Figures 4 and 5 clarify probability plots of the comparative models to show the over fitting of the MOEGM distribution.Figure 3
(a) Histogram and estimated p.d.f., (b) Estimated c.d.f. and the empirical c.d.f., (c) generate qunatile and data: flood peaks data.
(a)
(b)
(c)Figure 4
PP plot for different models.Figure 5
PP plot for different models.Based on the results in Table4 and Figures 3 and 4, we conclude that the MOEGM distribution is a better fit than comparative models for this data set.Table5 discussed MLE and Bayesian estimation methods comparing by SE, we note that the Bayesian estimation has smaller SE than MLE. The trace plots and the convergence plots of parameters by MCMC results of the MOEGM distribution are obtained in right and left Figure 6. The posterior density of MCMC findings for each parameter is shown in the center of Figure 6, which indicates a symmetric normal distribution comparable to the proposed distribution.Table 5
MLE and Bayesian estimation methods comparing by SE: flood peaks data.
MLE
Bayesian
Estimates
SE
Estimates
SE
θ
0.0141
0.0569
0.014041
0.003221
α
0.0185
0.0379
0.018479
0.001432
λ
0.0319
0.0382
0.031859
0.001461
γ
0.2966
0.3969
0.294978
0.148231Figure 6
The trace plots, posterior density and the convergence for parametersθ, α, λ and γ: flood peaks data.


### 7.2. Second Real Data of Stochastic Processes

In this subsection, we discuss data set of stochastic processes which was first introduced by Aarset [34] and represents the lifetimes of 50 devices (in weeks). This data set, also reported in Benkhelifa [30] BGG distribution, and Abd El-Bar [29] to discuss TGM distribution, is: “0.1, 0.2, 1, 1, 1, 67, 67, 67, 72, 75, 79, 1, 1, 2, 3, 6, 7, 11, 60, 63, 63, 12, 18, 18, 85, 85, 85, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 67, 82, 82, 83, 84, 84, 84, 85, 85, 86, 86.” The TGM distribution is better than Gompertz, shifted Gompertz, transmuted Lindley, Gompertz Makeham, transmuted Burr type III, transmuted Gompertz, transmuted exponentiated exponential, and transmuted generalized linear exponential distributions, for more details see Abd El-Bar [29]. The BGG distribution is better than Gompertz, beta generalized exponential, generalized exponential, beta Gompertz, exponential, beta exponential, and GG, for more details see Benkhelifa [30].Table6 presents the MLEs of the model parameters for the stochastic processes data set. The values of AICM, BICM, AICCM, HQICM, KSS and the PVKS are presented for the MOEGM model and the TGM, and the BGG distribution.Table 6
Comparison betweenMOEGM, BGG and TGM distributions:stochastic processes data.
θ
α
λ
γ
b
KSS
MOEGM
0.0096
1.593E − 10
0.2511
0.5168
0.1265
TGM
0.0998
0.0003
0.0115
0.1280
0.1517
BGG
0.0638
0.0029
1.3644
0.1198
0.1776
0.1266
CVMS
AICM
AICCM
BICM
HQICM
PVKS
MOEGM
0.1214
440.6824
441.5713
448.3305
443.5948
0.4003
TGM
0.2751
454.0590
454.9480
461.7070
456.9710
0.2002
BGG
0.157543
461.0024
462.366
470.5625
464.6429
0.3871From Table6, we conclude that the MOEGM model gives the best fit, where the values AICM, BICM, AICCM, HQICM, and KSS are smaller for MOEFM distribution than TGM and BGG distribution, and the PVKS is higher for the MOEGM model than TGM and BGG distribution. Figures 7(a), 7(b) illustrate the p.d.f.s, empirical c.d.f.s and probability plots, respectively, of the comparative models to show the over fitting of the MOEGM distribution.Figure 7
(a) Histogram and estimated p.d.f., (b) Estimated c.d.f. and the empirical c.d.f.: stochastic processes data.
(a)
(b)Table7 discussed MLE and Bayesian estimation methods comparing by SE, we note Bayesian estimation has smaller SE than MLE.Table 7
MLE and Bayesian estimation methods comparing by SE: stochastic processes data.
MLE
Bayesisna
Estimates
SE
Estimates
SE
θ
0.0096
0.3547
0.4195
0.2138
α
1.593E − 10
8.680E − 07
3.34E − 09
4.13E − 09
λ
0.2511
0.0945
0.2514
0.0338
γ
0.5168
0.0568
0.5015
0.0191


## 7.1. First Real Data of Flood Peaks

In this subsection, the first application of real data set is employed to illustrate the importance of theMOEGM distribution. This data set represents 72 excrescences of flood peaks for the years 1958–1984 (rounded to one decimal place) of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, Canada. The first data set is: “1.7, 2.5, 27.4, 1.0, 27.1, 2.2, 22.9, 1.7, 0.1, 1.1, 14.4, 1.1, 0.4, 20.6, 5.3, 0.7, 1.9, 13.0, 12.0, 9.3, 1.4, 18.7, 8.5, 25.5, 11.6, 21.5, 27.6, 36.4, 2.7, 14.1, 22.1, 1.1, 2.5, 14.4, 1.7, 37.6, 0.6, 2.2, 39.0, 0.3, 15.0, 11.0, 7.3, 0.6, 9.0, 1.7, 7.0, 20.1, 0.4, 2.8, 14.1, 9.9, 10.4, 10.7, 30.0, 3.6, 5.6, 30.8, 13.3, 4.2, 25.5, 3.4, 11.9, 64.0, 1.5, 20.2, 16.8, 5.3, 9.7, 27.5, 2.5 and 7.0.” The fit of the proposed model are compared with the transmuted Gompertz-Makeham (TGM) (Abd El-Bar [29]), beta generalized Gompertz (BGG) (Benkhelifa [30]), kumaraswamy gompertz makeham (KGM) (Chukwu and Ogunde [5]), Gompertz Lomax (GL) (Oguntunde et al. [31]), exponentiated generalized Weibull-Gompertz (EGWG) (El-Bassiouny et al. [32]), generalized Gompertz (GG) (El-Gohary et al. [33]) and Gompertz models.Table4 presents the MLEs with standard error (SE) of the model parameters for the first data set. The values of AICM, BICM, AICCM, HQICM, KSS and the PVKS are presented for the MOEGM model and the other models.Table 4
MLEs of the models parameters with SE and different measures for fitting: flood peaks data.
MLEs
SE
KSS
PVKS
CVMS
ADS
AICM
AICCM
BICM
HQICM
MOEGM
θ
0.0141
0.0569
0.0985
0.4873
0.1014
0.5756
506.9077
507.5047
516.0144
510.5331
α
0.0185
0.0379
λ
0.0319
0.0382
γ
0.2966
0.3969
GL
θ
1.7323
3.7299
0.0999
0.4691
0.1193
0.6709
507.6678
508.2648
516.7744
511.2931
α
0.4989
3.6786
λ
1.2268
5.6233
γ
0.1877
0.6431
Gompertz
λ
0.0013
0.0081
0.1463
0.0918
0.1120
0.6459
508.2565
508.429
518.8094
512.8534
α
0.0852
0.0148
EGWG
θ
1.6924
19.7580
0.1026
0.4351
0.1027
0.6039
509.3512
510.2603
520.7345
513.8829
λ
0.4178
4.5617
α
0.1118
4.4805
γ
0.5656
3.0847
β
0.3331
0.4674
BGG
λ
0.0123
0.0191
0.0998
0.4799
0.4754
0.5876
508.5600
509.4691
519.9433
513.0917
α
0.0331
0.0851
γ
0.5632
1.6298
θ
1.4283
4.6792
β
1.6742
3.1784
KGM
λ
0.0229
0.0442
0.1034
0.4251
0.1102
0.5928
508.7930
509.7020
520.1763
513.3247
α
0.0160
0.0508
γ
0.0399
0.0788
a
0.7910
0.1410
b
1.1351
0.9661
TGM
λ
0.0036
0.0596
0.1290
0.1821
0.1112
0.6376
508.4915
509.0885
517.5982
512.1169
α
0.0655
0.0499
θ
0.0053
0.0060
γ
0.2544
0.4056From Table4, we conclude that the MOEGM model gives the best fit, where the values of AICM, BICM, AICCM, HQICM, and KSS are smaller and the PVKS is higher for the MOEGM model when compared with those values of the other models. Figures 3(a), 3(b) illustrate the p.d.f., empirical c.d.f.s and probability plots, respectively, of the comparative models to show the over fitting of the MOEGM distribution. Figures 3(a)–3(c) illustrate estimated p.d.f. with histogram, estimated c.d.f. with empirical c.d.f., and Q-Q plot of the MOEGM distribution, respectively. Figures 4 and 5 clarify probability plots of the comparative models to show the over fitting of the MOEGM distribution.Figure 3
(a) Histogram and estimated p.d.f., (b) Estimated c.d.f. and the empirical c.d.f., (c) generate qunatile and data: flood peaks data.
(a)
(b)
(c)Figure 4
PP plot for different models.Figure 5
PP plot for different models.Based on the results in Table4 and Figures 3 and 4, we conclude that the MOEGM distribution is a better fit than comparative models for this data set.Table5 discussed MLE and Bayesian estimation methods comparing by SE, we note that the Bayesian estimation has smaller SE than MLE. The trace plots and the convergence plots of parameters by MCMC results of the MOEGM distribution are obtained in right and left Figure 6. The posterior density of MCMC findings for each parameter is shown in the center of Figure 6, which indicates a symmetric normal distribution comparable to the proposed distribution.Table 5
MLE and Bayesian estimation methods comparing by SE: flood peaks data.
MLE
Bayesian
Estimates
SE
Estimates
SE
θ
0.0141
0.0569
0.014041
0.003221
α
0.0185
0.0379
0.018479
0.001432
λ
0.0319
0.0382
0.031859
0.001461
γ
0.2966
0.3969
0.294978
0.148231Figure 6
The trace plots, posterior density and the convergence for parametersθ, α, λ and γ: flood peaks data.


## 7.2. Second Real Data of Stochastic Processes

In this subsection, we discuss data set of stochastic processes which was first introduced by Aarset [34] and represents the lifetimes of 50 devices (in weeks). This data set, also reported in Benkhelifa [30] BGG distribution, and Abd El-Bar [29] to discuss TGM distribution, is: “0.1, 0.2, 1, 1, 1, 67, 67, 67, 72, 75, 79, 1, 1, 2, 3, 6, 7, 11, 60, 63, 63, 12, 18, 18, 85, 85, 85, 18, 18, 18, 21, 32, 36, 40, 45, 46, 47, 50, 55, 67, 82, 82, 83, 84, 84, 84, 85, 85, 86, 86.” The TGM distribution is better than Gompertz, shifted Gompertz, transmuted Lindley, Gompertz Makeham, transmuted Burr type III, transmuted Gompertz, transmuted exponentiated exponential, and transmuted generalized linear exponential distributions, for more details see Abd El-Bar [29]. The BGG distribution is better than Gompertz, beta generalized exponential, generalized exponential, beta Gompertz, exponential, beta exponential, and GG, for more details see Benkhelifa [30].Table6 presents the MLEs of the model parameters for the stochastic processes data set. The values of AICM, BICM, AICCM, HQICM, KSS and the PVKS are presented for the MOEGM model and the TGM, and the BGG distribution.Table 6
Comparison betweenMOEGM, BGG and TGM distributions:stochastic processes data.
θ
α
λ
γ
b
KSS
MOEGM
0.0096
1.593E − 10
0.2511
0.5168
0.1265
TGM
0.0998
0.0003
0.0115
0.1280
0.1517
BGG
0.0638
0.0029
1.3644
0.1198
0.1776
0.1266
CVMS
AICM
AICCM
BICM
HQICM
PVKS
MOEGM
0.1214
440.6824
441.5713
448.3305
443.5948
0.4003
TGM
0.2751
454.0590
454.9480
461.7070
456.9710
0.2002
BGG
0.157543
461.0024
462.366
470.5625
464.6429
0.3871From Table6, we conclude that the MOEGM model gives the best fit, where the values AICM, BICM, AICCM, HQICM, and KSS are smaller for MOEFM distribution than TGM and BGG distribution, and the PVKS is higher for the MOEGM model than TGM and BGG distribution. Figures 7(a), 7(b) illustrate the p.d.f.s, empirical c.d.f.s and probability plots, respectively, of the comparative models to show the over fitting of the MOEGM distribution.Figure 7
(a) Histogram and estimated p.d.f., (b) Estimated c.d.f. and the empirical c.d.f.: stochastic processes data.
(a)
(b)Table7 discussed MLE and Bayesian estimation methods comparing by SE, we note Bayesian estimation has smaller SE than MLE.Table 7
MLE and Bayesian estimation methods comparing by SE: stochastic processes data.
MLE
Bayesisna
Estimates
SE
Estimates
SE
θ
0.0096
0.3547
0.4195
0.2138
α
1.593E − 10
8.680E − 07
3.34E − 09
4.13E − 09
λ
0.2511
0.0945
0.2514
0.0338
γ
0.5168
0.0568
0.5015
0.0191


## 8. Conclusion

Based on Marshall and Olkin approach, a new four-parameter extended Gompertz Makeham distribution was developed, the Marshall-Olkin extended Gompertz Makeham distribution. It includes special models, the Marshall-Olkin extended Makeham, Marshall-Olkin Gompertz Makeham, Gompertz Makeham, and Makeham distributions. Depending on the shape parameters, theMOEGM density function can take on a variety of shapes. Furthermore, depending on the design parameters, its hazard rate function might take on various shapes. We have included some statistical features. The method of likelihood and Bayesian estimation methods are used to estimate the unknown parameters of the proposed distribution. An MCMC technique is used to give a comparison for the estimated parameters. These comparisons were made using bias and MSE as criteria. The MSE and Bias of the Bayesian-based SELF are superior to both MLE in our simulation case. Real data sets were observed and it was noted that the MOEGM distribution resulted in the best fit. To summaries, the MOEGM distribution may provide a relatively flexible mechanism for fitting a wide range of positive real-world data sets. The novel distribution may be a feasible alternative to existing models now available in the literature for modeling actual data in domains like as engineering, survival analysis, hydrology, economics, and others.



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