# A New Useful Exponential Model with Applications to Quality Control and Actuarial Data

Authors: Aisha Fayomi; M. H. Tahir; Ali Algarni; M. Imran; Farrukh Jamal
Journal: Computational Intelligence and Neuroscience
 (2022)

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## Abstract

The compounding approach is used to introduce a new family of distributions calledexponentiated Bell G, analogy to exponentiated G Poisson. Several essential properties of the proposed family are obtained. The special model called exponentiated Bell exponential (EBellE) is presented along with properties. Furthermore, the risk theory related measures including value-at-risk and expected-shortfall are also computed for the special model. Group acceptance sampling plan is designed when a lifetime of a product or item follows an EBellE model taking median as a quality parameter. The parameters of the proposed model are estimated by considering maximum likelihood approach along with simulation analysis. The usefulness of the proposed model is illustrated by practical means which yield better fits as compared to several exponential related extended models.


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## Body

## 1. Introduction

Effective implementation of mathematical and statistical models enables the actuarial scientists to know as much as possible about future claims in a portfolio. These models serve as a guide to achieve better business and risk management decision and policies. Actuaries usually deal with a complex data such as right skewed, unimodal, and having heavy tail. The readers are referred to works of Klugman et al. [1], Cooray and Ananda et al. [2], Lane [3], Vernic [4], and Ibragimov et al. [5]. At the same time, they are eager on some flexible models which are capable of capturing the behaviours of such data to finding along with information when the real development deviates from the expected. The classical models are limited with their tail properties and goodness of fit tests. For instance, Pareto, Lomax, Fisk, and Dagum distribution are excessively used to model statistical size distributions in economics and actuarial sciences but often failed to provide better fits for many application. The Weibull distribution is appropriate for small losses but fail to uncover adequate trend, level, and trajectory for large losses [6]. The reader are referred to [7] for detail discussion on statistical size distributions which can be used in economics and actuarial sciences. To overcome the drawback of classical models, a substantial progress on persistent base related to distribution theory is documented in statistical literature. From the last couple of decades, the emerging trend has been seen in the generalization of the existing classical models. The models are extended by adopting different modes of adding one or more additional shape parameter(s) in the distribution. The basic aim of this whole exercise is to improve the tail properties as well as goodness of fit test of the classical models. There are several well-known generators which are documented in the statistical literature; the readers are referred to the works of Tahir and Nadarajah [8], Tahir and Cordeiro [9], Maurya and Nadarajah [10], and Lee et al. [11].Several new models related to claim data have recently been reported in statistical literature. Ahmad et al. [12] proposed a new method to define heavy-tailed distributions called the exponentiated power Weibull distribution with application to medical care insurance and vehicle insurance. Calderin–Ojeda and Kwok [13] presented a new class of composite model by using the Stoppa distribution and mode matching procedure and modelling the actuarial claims data of mixed sizes. Ahmad et al. [14] suggested nine new methods to define new distributions suitable for modelling heavy right-tail data with application to medical care insurance and vehicle insurance. Afify et al. [15] proposed a new heavy-tailed exponential distribution with application to unemployment claim data. Ahmad et al. [16] introduced a class of claim distributions useful in a number of lifetime analyses. A special submodel of the proposed family, called the Weibull claim model, is considered in detail with claim data application. Among classical discrete distributions, Poisson distribution is a most frequently used distribution for count data. Furthermore, it is extended into G-class and several transformation and family of distributions have been proposed. A detail review study on Poisson generated family of distributions, extensions, and transformation is recently presented by [10]. Castellares et al. [17] introduced a discrete Bell distribution from well-known Bell numbers, as a competitor or counterpart to Poisson distribution which exhibits many interesting properties such as a single parameter distribution, and it belongs to one-parameter exponential family of distributions and the Poisson distributions. They investigated that the Poisson model cannot be nested into the Bell model, but small values of the parameter the Bell model tends to Poisson distribution. Furthermore, the Bell model is infinity divisible and has larger variance as compared to the mean, which can be used to overcome the phenomenon of over-dispersion and zero-vertex for count data. The characteristics of the Bell model motivated us to develop a generalized class of distributions through compounding and to compare its mathematical and empirical characteristics with compounded Poisson-G class and its special models.The rest of the study is organized as follows. In Section2, we define the proposed EBell-G family of distributions. Section 3 provides the general mathematical and structural properties of EBell-G family of distributions including linear representation of density, quantile function, rth moments, probability weighted moments, analytical shapes of the density and hazard rate, entropy measures, reversed order statistics, upper record statistics, stochastic ordering, and parameters’ estimation by using maximum likelihood estimation. Section 4 illustrates the layout of the special model called EBellE as well as its essential properties, while Section 5 shows the commonly used actuarial measures, specially value-at-risk and expected-shortfall. Section 6 are illustrated group acceptance sampling plane when a lifetime of a certain product or item follows the EBellE model which is presented. The simulation analysis is presented in Section 7, and Section 8 contains the application of real datasets. The concluding remarks are given in Section 9.


## 2. Layout and Formulation of EBell-G Family

A single parameter discrete Bell distribution has been recently introduced by Castellares et al. [17], which is an analogy to discrete Poisson distribution but provides better fits compared to other discrete models including the Poisson model. The following expression given by Bell [18] is(1)expex−1=∑n=0∞Bnn!xn,where Bn denote the Bell numbers and can be derived from the following mathematical expression:(2)Bn=1e∑k=0∞knk!.(3)ℙrX=x=λne1−eλBnn!1−e1−eλ,x=1,2….Remark 1.
The Bell numberBn in (2) is the nth moment of the Poisson distribution with parameter equal to 1.
By considering equations (1) and (2), Castellares et al. [17] introduced a single-parameter Bell distribution defined by the following probability mass function (pmf) as(4)ℙrX=x=λxe−eλ+1Bxx!,x=0,1,2….Proposition 1.
LetXfollow a discrete Bell model with parameterλ; then, the following expression represents the pmf of Bell truncated model asWe first give the motivation for the proposed family. Suppose a system is havingN subsystems that are working or functioning independently at a given specific time. Here, Yi denotes the life of ith subsystem and θ parallel units constitutes the subsystem. Furthermore, the system will fail or remain functioning if all the subsystem fail; this is for the parallel system. On the contrary, for series system, the failure of any subsystem yields complete destruction of the whole system. Let us have a random variable rvN that follows any discrete distribution having pmf ℙN=n. Here, we suppose that a component Zi,1,……Zi,θ having failure time for the ith subsystem are i.i.d. with suitable cdf depending upon the vector τ, say for X>0, TGx,τ,θ=Gx,τθ. If we define Y=minY1,……YN, then the conditional cdf of Y given N is as follows:(5)FY|N=PminY1,…,YN<y|N=1−1−PθZ1,1≤yN=1−1−Gy;τθN.The unconditional cdf ofY corresponding to (5) is given by(6)Fy=∑n=1∞Fy|NPN=n=1−1−Gy;τθNPN=n.By using the Bell truncated model given in Eq. (4) and then using Eq. (6), the unconditional cdf of X is defined below as follows.(7)Fx=1−exp−eλ1−e−λGθx1−exp1−eλ,(8)fx=λθgxGθ−1xexpλ1−Gθxexp−eλ1−e−λGθx1−exp1−eλ,(9)sx=exp−eλ1−e−λGθx−exp1−eλ1−exp1−eλ,hx=λθgxGθ−1xexpλ1−Gθxexp−eλ1−e−λGθxexp−eλ1−e−λGθx−exp1−eλ.Proposition 2.
LetX∼EBell-Gλ,θ,ξ, forx>0andλ,θ,ξ>0; then, its cumulative distribution function (cdf) having baseline pdf and cdf respectivelygxandGxis given byProposition 3.
LetX∼EBell-Gλ,θ,ξforx>0andλ,θ,ξ>0; then, its probability distribution function (pdf) having Eq. (8), with baseline pdf and cdf respectivelygxandGx, is given byProposition 4.
LetX∼EBell-Gλ,θ,ξforx>0andλ,θ,ξ>0; then, its survival function (sf) and hazard rate function (hrf) are, respectively, given by


## 3. Properties of the EBell-G Family

This section provides some mathematical properties of the EBell-G family of distributions.


### 3.1. Quantile Function

Quantile function (qf) is an important measure for generating random numbers and several other important uses in quality control sampling plans and in risk theory; the two important commonly used measures value-at-risk (VaR) and expected-shortfall (ES) which depend on qf and is given as follows.(10)Qu=F−11−λ−1loglog1−u1−exp1−eλ+expλ1/θ.Proposition 5.
LetX∼EBell-Gλ,θ,ξforx>0andλ,θ,ξ>0; then, the expression of qf is given below, whereu∼uniform0,1, and by replacingu=0.5, it yields the median of the EBell-G:


### 3.2. Analytic Shapes of the Density and Hazard Rate Function

The analytical shapes of the density and hrf can be yielded for EBellE, respectively, as follows:(11)g′xgx+θ−1gxGx−λθgxGθ−1x1+expλ1−Gθx=0,g′xgx+θ−1gxGx−λθgxGθ−1x=0.


### 3.3. Useful Expansions

Here, we show the useful expansion for EBell-G density can be used to drive several important properties by taking into account the following two series to obtain the expansion for EBell-G.(12)1−tb=∑c=0∞−1cbctc.Proposition 6.
The generalized binomial expansion which holds for any real noninteger b andt<1isThe power series for exponential function is given by Bourguignon et al. [19] and is given as follows:(13)exp−αxb=∑k=0∞−1kαkxkbk!.Therefore, by using Eq. (11) to Eq. (8), we can deduce pdf and cdf, simultaneously, as(14)fx;λ,θ,ξ=∑v=0∞wvhθv+1x,(15)Fx;λ,θ,ξ=∑v=0∞wvHθv+1x,where(16)wv=λ1+vv+1−11−exp1−eλv!∑k∞∑r=0∞1k!−1k+r+vkr1+rveλ1+k,are constants satisfying ∑v=0∞wv=1. Eq. (12) represents exp-G, that is, hθv+1x and the term θv+1 is treated as the power parameter. By using Eq. (12), numerous properties of G-class can be obtained.


### 3.4. Mathematical Properties

One can derive some important mathematical properties by considering Eq. (12). The rth raw moment of X is given by(17)μr′=∑v=0∞wvEXθv+1r,where EXθv+1r follows a exp-G with θv+1 treated as the power parameter, and by taking r=1, in (14), yields the mean for X.The incomplete moments are important and have many practical uses. The expression ofsth incomplete moments, denoted by φst, is defined by φst=∫−∞txsfxdx and can be obtained by using Eq. (12) as(18)φst=∑v=0∞wv∫−∞txshθv+1xdx.The first incomplete moment of the EBell-G family can be obtained as by takings=1 in Eq. (15). The sth incomplete moment is an important to compute several measures, namely, mean deviations from mean and median, mean waiting time, conditional moments, and income inequality measures among others.


### 3.5. Probability Weighted Moments

Thes,rth probability weighted moments (PWM) of X following the EBell-G family, say ρs,r, is formally defined by(19)ρs,r=EXsFxr=∫−∞+∞XsFxrfxdx.By using Eq. (7) and Eq. (8), we can obtain(20)ρs,r=∑Q=0∞wQEYθQ+1s,where(21)wQ=−1Qλ1+QeλQ!Q+11−exp1−eλr+1∑z,v,p=0∞−1z+v+przvp1+zeλv1+pQv!.


### 3.6. Entropy Measures

The entropy measures are important to underline the randomness or uncertainty or diversity of the system. The most frequently used index of dispersion in ecology as well as in statistics is called the Rényi entropyIδx and is defined by the following expression:(22)Iδx=1−δ−1log∫−∞+∞fxδdx,where δ>0 and δ≠1, which then follows(23)IδX=1−δ−1log∑b=0∞Qb∫−∞+∞gxδGxθb+δθ−1dx,where(24)Qb=θδλδ+beδλb!1−exp1−eλδ∑t,s=0∞1t!−1t+s+btsδeλts+δb.The Shannon entropy say,Hqx, can be obtained by the following expression:(25)Hqx=1−q−1log∑b=0∞Qb∫−∞+∞gxqGxbdx,where q>0 and q≠1 and(26)Qb=θqλq+beqλb!1−exp1−eλq∑t,s=0∞1t!−1t+s+btsqets+qb.


### 3.7. Order Statistics

Here, we derived the explicit expression for theith-order statistics for EBell-G, say fi:nx. Let a sample of size be n; then, the pdf of ith-order statistics is defined by(27)fi:nx=1Bi,n−i+1fx∑l=0n−i−1ln−ilFxi+l−1.By using Eq. (7) and Eq. (8), the density for EBell-G can be written as(28)fi:nx=∑j=0∞Qi:njhθj+1x,where(29)Qi:nj=λ1+j1+j−1−1jeλj!1−exp1−eλi+l1Bi,n−i+1∑l=0n−i∑p=0i+l−1∑z,i∞eλzz!−1p+z+i+l×i+l−1pn−ilzi1+pz1+ij.Thesth moment of order statistic can be obtained as(30)EXi:ns=∑j=0∞Qi:njμθj+1s,where μθj+1s is the sth moment of Exp-G distribution with power parameter θj+1.


### 3.8. Reversed Order Statistics

The reversed order statistics can be used whenx1,……,xn are arranged in the decreasing order; for more detail, see the work of Jamal et al. [20]. The pdf of Xrre:n, represented by frre:nx=fn−r+1:nx, is defined by(31)frre:nx=Cr:nfxFxn−r1−Fxr−1,x∈ℝ,and (32)frre:nx=Cr:nfx∑l=0r−1−1lr−1lFxn−r+l.Consider(33)I=fx1−exp−eλ1−e−λGθxn−r+l1−exp1−eλn−r+l.By using Eq. (10), we can obtain(34)I=fx1−exp1−eλn−r+l∑p=0n−r+l−1pn−r+lpexp−peλ1−e−λGθx.Then, by using Eq. (11), we can have(35)I=λθgxGθ−1x1−exp1−eλn−r+l+1expλ1−Gθx∑p=0n−r+l−1pn−r+lp×exp−eλ1−e−λGθx1+p.Let us consider(36)exp−eλ1−e−λGθx1+p=∑z=0∞−1zz!eλ1+pz1−e−λGθxz.After simplification, we have shapes:(37)I=λθgxGθ−1x1−exp1−eλn−r+l+1expλ1−Gθx∑p=0n−r+l∑z=0∞∑z=0∞−1p+z+kn−r+lp×zkeλ1+pzz!e−λkGθx,I=λθgxGθ−1xeλ1−exp1−eλn−r+l+1∑p=0n−r+l∑z=0∞∑k=0∞−1p+z+kn−r+lpzk×eλ1+pzz!e−λ1+kGθx,Finally,(38)I=λeλ1−exp1−eλn−r+l+1∑p=0n−r+l∑z=0∞∑k=0∞∑j=0∞−1p+z+k+jn−r+lpzkeλ1+pzz!j!j+1×λ1+kjθj+1gxGxθj+1−1.The reduced form will be(39)frre:nx=∑j=0∞Wr:njhθj+1x,where hθj+1=θj+1gxGxθj+1−1 and(40)Wr:nj=λeλ1−exp1−eλn−r+l+1Cr:n∑l=0r−1∑p=0n−r+l∑z=0∞∑k=0∞−1p+z+k+j+ln−r+lpr−1l×zkeλ1+pzz!j!j+1λ1+kj.Thepth moment of reversed-order statistic can be obtained as(41)EXr:np=∑j=0∞Wi:njμθj+1p,where μθj+1p is the pth moment of Exp-G distribution with power parameter θj+1.


### 3.9. Upper Record Statistics

Record value is an important measure in many practical areas, for instance, economics data and weather and athletic events. Let us considerXnn≥1 a sequence of independent rvs having the same distribution. Let us denote by Fx and fx the related cdf and pdf of EBellE distribution, respectively, and Xi:n be the ith-order statistic as described previously. For fixed k≥1, the pdf of kth upper record statistic is defined by(42)fYnkx=k!n−1!Rxn−11−Fxk−1fx,where Rx=−ln1−Fx correspond to the cumulative hazard rate function related to Fx. Eq. (20) can also be expressed for Rx=eλ1−e−λGθx, by using (7), as(43)fYnkx=k!eλn−1n−1!∑t=0k−1−1tk−1t1−e−λGxn−1fxFxt.Considering the last terms,(44)I=1−e−λGθxn−1fxFxtand after using series, we obtain(45)I=1−e−λGθxn−1λθgxGθ−1xexpλ1−Gθx1−exp1−eλt+1∑z=0∞−1ztz×exp−eλ1−e−λGθxz+1,Using power series given in Eq. (11), we obtain(46)exp−eλ1−e−λGθx1+z=∑v=0∞−1vv!eλ1+zv1−e−λGθxv.Now, the above expression becomes(47)I=λθgxGθ−1xexpλ1−Gθx1−exp1−eλt+1∑z=0∞∑v=0∞−1z+vv!tzeλ1+zv×1−e−λGθxv+n−1.By using Eq. (11) again, we obtain(48)I=λθgxGθ−1xeλ1−exp1−eλt+1∑z=0∞∑v=0∞∑p=0∞−1z+v+pv!tzv+n−1p×eλ1+zve−λ1+pGθx.Finally, we have(49)I=λeλq+1−11−exp1−eλt+1∑q=0∞∑z=0∞∑v=0∞∑p=0∞−1z+v+p+qv!q!tzv+n−1peλ1+zv×λ1+pqθq+1gxGθq+1−1x.The reduced form becomes(50)fYnkx=∑q=0∞Wqhθq+1x,where hθq+1x=θq+1gxGθq+1−1x and(51)Wq=k!eλnλq+1−1n−1!∑t=0k−1∑z,v,p=0∞−1z+v+p+q+tv!q!1−exp1−eλt+1tzv+n−1p×k−1teλ1+zvλ1+pq.A random sample of 50 is generated from the EBellE model using Eq. (23), and then, take k=3 and α=β=λ=0.5. Table 1 shows a random sample of 50 from the EBellE model along with upper XUn and lower XLn records values. The plot of lower and upper record values is illustrated in Figure 1. The Records package is used in R-Statistical Computing Environment to compute XUn and XLn records’ values.Table 1
Upper and lower record values from EBellE generated data.
n=50; k=3; α=θ=λ=0.5XUnXLn0.2916810.6087761.8723130.3385390.0649550.0832340.2916810.0832340.3997180.5082710.006260.0428020.1852900.185290.185290.0094711.1325110.0222770.0001410.2916810.0832340.7475750.4611160.1180132.3589660.8942750.5094820.0644430.0052020.1200050.1631450.1806310.0520730.6087760.0632570.0413760.1873930.7996980.9835060.0199170.7475750.0413760.0644430.0018811.5991341.4632320.0001971.1325110.0094710.0632570.0590460.0473853.7578720.2597161.5794190.006260.5094820.1561731.1071950.7196950.0077131.5991340.0052020.1883531.5794190.1182760.2809221.3030781.8723130.001881Figure 1
Plot of upper (a) and lower (b) record values of the EBellE model at some parametric values.
(a)(b)


### 3.10. Stochastic Ordering

Stochastic ordering is another important tool in statistics to define the comparative behaviour specifically in reliability theory. Suppose the tworvs, say X1 and X2 and under specific circumstance; let us consider that rvX1 is lower than X2; the readers can refer to the work of Khan et al. [21] for detailed illustration on four stochastic ordering and their well-established relationships.(52)ddxlogf1xf2x=θgxGθ−1x;ξλ1−λ2+λ1eλ1Gθx;ξ−λ2eλ2Gθx;ξ<0.Theorem 1.
LetX1∼EBell-Gλ1,θ;ξandX2∼EBell-Gλ2,θ;ξ. Ifα1≤α2, thenX1≤lrX2:Proof. First, we have the ratio(53)f1xf2x=λ1expλ11−Gθxexp−eλ11−e−λ1Gθx1−exp1−eλ1−1λ2expλ21−Gθxexp−eλ21−e−λ2Gθx1−exp1−eλ2−1.Now, consider(54)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+θgxGθ−1xλ2eλ21−Gθx−λ1eλ11−Gθx.After simplification, we obtain(55)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+λ2eλ21−Gθx−λ1eλ11−Gθx.Ifλ1<λ2, we obtain(56)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+λ2eλ21−Gθx−λ1eλ11−Gθx<0.Thus,f1x/f2x is decreasing in x, and hence, X1≤lrX2. This completes the proof.


### 3.11. Estimation of Family Parameters

This section is about estimation of the unknown parameters estimation of the EBell-G model by taking into account the popular estimation method known as maximum likelihood estimation (MLE). There are several advantages of MLE over other estimation methods; for instance, the maximum likelihood estimates fulfil the required properties that can be used in constructing confidence intervals as well as maximum likelihood estimates delivering simple approximation very handy while working the finite sample.ℓ. represent the vector parameters ϕ=λ,θ,ξ⊤; then,(57)ℓϕ=nlogλθ+∑i=1∞loggxi;ξ+θ−1∑i=1∞logGxi;ξ+λ∑i=1∞1−Gθxi;ξ−nexpλ+∑i=1∞expλ1−Gθxi;ξ−nlog1−exp1−eλ,(58)∂ℓ∂λ=nλ+∑i=1∞1−Gθxi;ξ−nexpλ+∑i=1∞expλ1−Gθxi;ξ1−Gθxi;ξ−nexpλexp1−eλ1−exp1−eλ,∂ℓ∂θ=nθ+∑i=1∞logGxi;ξ−λ∑i=1∞Gθxi;ξlogGxi;ξ−λ∑i=1∞expλ1−Gθxi;ξGθxi;ξlogGxi;ξ,∂ℓ∂ξ=∑i=1∞giξgxi;ξ+θ−1∑i=1∞GiξGxi;ξ−λθ∑i=1∞Gθ−1xi;ξGiξ−λθ∑i=1∞expλ1−Gθxi;ξGθ−1xi;ξGiξ,where giξ=∂/∂ξgxi;ξ and Giξ=∂/∂ξGxi;ξ are derivatives of column vectors of the same dimension of ξ, and by setting ϕλ=0,ϕθ=0, and ϕξ=0, the MLEs can be yielded by solving the above equations simultaneously.Proposition 7.
A randomly selected sample of sizenis under EBell-G; then, the score vectorϕλ,ϕθ,ϕξis given by


## 3.1. Quantile Function

Quantile function (qf) is an important measure for generating random numbers and several other important uses in quality control sampling plans and in risk theory; the two important commonly used measures value-at-risk (VaR) and expected-shortfall (ES) which depend on qf and is given as follows.(10)Qu=F−11−λ−1loglog1−u1−exp1−eλ+expλ1/θ.Proposition 5.
LetX∼EBell-Gλ,θ,ξforx>0andλ,θ,ξ>0; then, the expression of qf is given below, whereu∼uniform0,1, and by replacingu=0.5, it yields the median of the EBell-G:


## 3.2. Analytic Shapes of the Density and Hazard Rate Function

The analytical shapes of the density and hrf can be yielded for EBellE, respectively, as follows:(11)g′xgx+θ−1gxGx−λθgxGθ−1x1+expλ1−Gθx=0,g′xgx+θ−1gxGx−λθgxGθ−1x=0.


## 3.3. Useful Expansions

Here, we show the useful expansion for EBell-G density can be used to drive several important properties by taking into account the following two series to obtain the expansion for EBell-G.(12)1−tb=∑c=0∞−1cbctc.Proposition 6.
The generalized binomial expansion which holds for any real noninteger b andt<1isThe power series for exponential function is given by Bourguignon et al. [19] and is given as follows:(13)exp−αxb=∑k=0∞−1kαkxkbk!.Therefore, by using Eq. (11) to Eq. (8), we can deduce pdf and cdf, simultaneously, as(14)fx;λ,θ,ξ=∑v=0∞wvhθv+1x,(15)Fx;λ,θ,ξ=∑v=0∞wvHθv+1x,where(16)wv=λ1+vv+1−11−exp1−eλv!∑k∞∑r=0∞1k!−1k+r+vkr1+rveλ1+k,are constants satisfying ∑v=0∞wv=1. Eq. (12) represents exp-G, that is, hθv+1x and the term θv+1 is treated as the power parameter. By using Eq. (12), numerous properties of G-class can be obtained.


## 3.4. Mathematical Properties

One can derive some important mathematical properties by considering Eq. (12). The rth raw moment of X is given by(17)μr′=∑v=0∞wvEXθv+1r,where EXθv+1r follows a exp-G with θv+1 treated as the power parameter, and by taking r=1, in (14), yields the mean for X.The incomplete moments are important and have many practical uses. The expression ofsth incomplete moments, denoted by φst, is defined by φst=∫−∞txsfxdx and can be obtained by using Eq. (12) as(18)φst=∑v=0∞wv∫−∞txshθv+1xdx.The first incomplete moment of the EBell-G family can be obtained as by takings=1 in Eq. (15). The sth incomplete moment is an important to compute several measures, namely, mean deviations from mean and median, mean waiting time, conditional moments, and income inequality measures among others.


## 3.5. Probability Weighted Moments

Thes,rth probability weighted moments (PWM) of X following the EBell-G family, say ρs,r, is formally defined by(19)ρs,r=EXsFxr=∫−∞+∞XsFxrfxdx.By using Eq. (7) and Eq. (8), we can obtain(20)ρs,r=∑Q=0∞wQEYθQ+1s,where(21)wQ=−1Qλ1+QeλQ!Q+11−exp1−eλr+1∑z,v,p=0∞−1z+v+przvp1+zeλv1+pQv!.


## 3.6. Entropy Measures

The entropy measures are important to underline the randomness or uncertainty or diversity of the system. The most frequently used index of dispersion in ecology as well as in statistics is called the Rényi entropyIδx and is defined by the following expression:(22)Iδx=1−δ−1log∫−∞+∞fxδdx,where δ>0 and δ≠1, which then follows(23)IδX=1−δ−1log∑b=0∞Qb∫−∞+∞gxδGxθb+δθ−1dx,where(24)Qb=θδλδ+beδλb!1−exp1−eλδ∑t,s=0∞1t!−1t+s+btsδeλts+δb.The Shannon entropy say,Hqx, can be obtained by the following expression:(25)Hqx=1−q−1log∑b=0∞Qb∫−∞+∞gxqGxbdx,where q>0 and q≠1 and(26)Qb=θqλq+beqλb!1−exp1−eλq∑t,s=0∞1t!−1t+s+btsqets+qb.


## 3.7. Order Statistics

Here, we derived the explicit expression for theith-order statistics for EBell-G, say fi:nx. Let a sample of size be n; then, the pdf of ith-order statistics is defined by(27)fi:nx=1Bi,n−i+1fx∑l=0n−i−1ln−ilFxi+l−1.By using Eq. (7) and Eq. (8), the density for EBell-G can be written as(28)fi:nx=∑j=0∞Qi:njhθj+1x,where(29)Qi:nj=λ1+j1+j−1−1jeλj!1−exp1−eλi+l1Bi,n−i+1∑l=0n−i∑p=0i+l−1∑z,i∞eλzz!−1p+z+i+l×i+l−1pn−ilzi1+pz1+ij.Thesth moment of order statistic can be obtained as(30)EXi:ns=∑j=0∞Qi:njμθj+1s,where μθj+1s is the sth moment of Exp-G distribution with power parameter θj+1.


## 3.8. Reversed Order Statistics

The reversed order statistics can be used whenx1,……,xn are arranged in the decreasing order; for more detail, see the work of Jamal et al. [20]. The pdf of Xrre:n, represented by frre:nx=fn−r+1:nx, is defined by(31)frre:nx=Cr:nfxFxn−r1−Fxr−1,x∈ℝ,and (32)frre:nx=Cr:nfx∑l=0r−1−1lr−1lFxn−r+l.Consider(33)I=fx1−exp−eλ1−e−λGθxn−r+l1−exp1−eλn−r+l.By using Eq. (10), we can obtain(34)I=fx1−exp1−eλn−r+l∑p=0n−r+l−1pn−r+lpexp−peλ1−e−λGθx.Then, by using Eq. (11), we can have(35)I=λθgxGθ−1x1−exp1−eλn−r+l+1expλ1−Gθx∑p=0n−r+l−1pn−r+lp×exp−eλ1−e−λGθx1+p.Let us consider(36)exp−eλ1−e−λGθx1+p=∑z=0∞−1zz!eλ1+pz1−e−λGθxz.After simplification, we have shapes:(37)I=λθgxGθ−1x1−exp1−eλn−r+l+1expλ1−Gθx∑p=0n−r+l∑z=0∞∑z=0∞−1p+z+kn−r+lp×zkeλ1+pzz!e−λkGθx,I=λθgxGθ−1xeλ1−exp1−eλn−r+l+1∑p=0n−r+l∑z=0∞∑k=0∞−1p+z+kn−r+lpzk×eλ1+pzz!e−λ1+kGθx,Finally,(38)I=λeλ1−exp1−eλn−r+l+1∑p=0n−r+l∑z=0∞∑k=0∞∑j=0∞−1p+z+k+jn−r+lpzkeλ1+pzz!j!j+1×λ1+kjθj+1gxGxθj+1−1.The reduced form will be(39)frre:nx=∑j=0∞Wr:njhθj+1x,where hθj+1=θj+1gxGxθj+1−1 and(40)Wr:nj=λeλ1−exp1−eλn−r+l+1Cr:n∑l=0r−1∑p=0n−r+l∑z=0∞∑k=0∞−1p+z+k+j+ln−r+lpr−1l×zkeλ1+pzz!j!j+1λ1+kj.Thepth moment of reversed-order statistic can be obtained as(41)EXr:np=∑j=0∞Wi:njμθj+1p,where μθj+1p is the pth moment of Exp-G distribution with power parameter θj+1.


## 3.9. Upper Record Statistics

Record value is an important measure in many practical areas, for instance, economics data and weather and athletic events. Let us considerXnn≥1 a sequence of independent rvs having the same distribution. Let us denote by Fx and fx the related cdf and pdf of EBellE distribution, respectively, and Xi:n be the ith-order statistic as described previously. For fixed k≥1, the pdf of kth upper record statistic is defined by(42)fYnkx=k!n−1!Rxn−11−Fxk−1fx,where Rx=−ln1−Fx correspond to the cumulative hazard rate function related to Fx. Eq. (20) can also be expressed for Rx=eλ1−e−λGθx, by using (7), as(43)fYnkx=k!eλn−1n−1!∑t=0k−1−1tk−1t1−e−λGxn−1fxFxt.Considering the last terms,(44)I=1−e−λGθxn−1fxFxtand after using series, we obtain(45)I=1−e−λGθxn−1λθgxGθ−1xexpλ1−Gθx1−exp1−eλt+1∑z=0∞−1ztz×exp−eλ1−e−λGθxz+1,Using power series given in Eq. (11), we obtain(46)exp−eλ1−e−λGθx1+z=∑v=0∞−1vv!eλ1+zv1−e−λGθxv.Now, the above expression becomes(47)I=λθgxGθ−1xexpλ1−Gθx1−exp1−eλt+1∑z=0∞∑v=0∞−1z+vv!tzeλ1+zv×1−e−λGθxv+n−1.By using Eq. (11) again, we obtain(48)I=λθgxGθ−1xeλ1−exp1−eλt+1∑z=0∞∑v=0∞∑p=0∞−1z+v+pv!tzv+n−1p×eλ1+zve−λ1+pGθx.Finally, we have(49)I=λeλq+1−11−exp1−eλt+1∑q=0∞∑z=0∞∑v=0∞∑p=0∞−1z+v+p+qv!q!tzv+n−1peλ1+zv×λ1+pqθq+1gxGθq+1−1x.The reduced form becomes(50)fYnkx=∑q=0∞Wqhθq+1x,where hθq+1x=θq+1gxGθq+1−1x and(51)Wq=k!eλnλq+1−1n−1!∑t=0k−1∑z,v,p=0∞−1z+v+p+q+tv!q!1−exp1−eλt+1tzv+n−1p×k−1teλ1+zvλ1+pq.A random sample of 50 is generated from the EBellE model using Eq. (23), and then, take k=3 and α=β=λ=0.5. Table 1 shows a random sample of 50 from the EBellE model along with upper XUn and lower XLn records values. The plot of lower and upper record values is illustrated in Figure 1. The Records package is used in R-Statistical Computing Environment to compute XUn and XLn records’ values.Table 1
Upper and lower record values from EBellE generated data.
n=50; k=3; α=θ=λ=0.5XUnXLn0.2916810.6087761.8723130.3385390.0649550.0832340.2916810.0832340.3997180.5082710.006260.0428020.1852900.185290.185290.0094711.1325110.0222770.0001410.2916810.0832340.7475750.4611160.1180132.3589660.8942750.5094820.0644430.0052020.1200050.1631450.1806310.0520730.6087760.0632570.0413760.1873930.7996980.9835060.0199170.7475750.0413760.0644430.0018811.5991341.4632320.0001971.1325110.0094710.0632570.0590460.0473853.7578720.2597161.5794190.006260.5094820.1561731.1071950.7196950.0077131.5991340.0052020.1883531.5794190.1182760.2809221.3030781.8723130.001881Figure 1
Plot of upper (a) and lower (b) record values of the EBellE model at some parametric values.
(a)(b)


## 3.10. Stochastic Ordering

Stochastic ordering is another important tool in statistics to define the comparative behaviour specifically in reliability theory. Suppose the tworvs, say X1 and X2 and under specific circumstance; let us consider that rvX1 is lower than X2; the readers can refer to the work of Khan et al. [21] for detailed illustration on four stochastic ordering and their well-established relationships.(52)ddxlogf1xf2x=θgxGθ−1x;ξλ1−λ2+λ1eλ1Gθx;ξ−λ2eλ2Gθx;ξ<0.Theorem 1.
LetX1∼EBell-Gλ1,θ;ξandX2∼EBell-Gλ2,θ;ξ. Ifα1≤α2, thenX1≤lrX2:Proof. First, we have the ratio(53)f1xf2x=λ1expλ11−Gθxexp−eλ11−e−λ1Gθx1−exp1−eλ1−1λ2expλ21−Gθxexp−eλ21−e−λ2Gθx1−exp1−eλ2−1.Now, consider(54)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+θgxGθ−1xλ2eλ21−Gθx−λ1eλ11−Gθx.After simplification, we obtain(55)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+λ2eλ21−Gθx−λ1eλ11−Gθx.Ifλ1<λ2, we obtain(56)ddxlogf1xf2x=θgxGθ−1xλ2−λ1+λ2eλ21−Gθx−λ1eλ11−Gθx<0.Thus,f1x/f2x is decreasing in x, and hence, X1≤lrX2. This completes the proof.


## 3.11. Estimation of Family Parameters

This section is about estimation of the unknown parameters estimation of the EBell-G model by taking into account the popular estimation method known as maximum likelihood estimation (MLE). There are several advantages of MLE over other estimation methods; for instance, the maximum likelihood estimates fulfil the required properties that can be used in constructing confidence intervals as well as maximum likelihood estimates delivering simple approximation very handy while working the finite sample.ℓ. represent the vector parameters ϕ=λ,θ,ξ⊤; then,(57)ℓϕ=nlogλθ+∑i=1∞loggxi;ξ+θ−1∑i=1∞logGxi;ξ+λ∑i=1∞1−Gθxi;ξ−nexpλ+∑i=1∞expλ1−Gθxi;ξ−nlog1−exp1−eλ,(58)∂ℓ∂λ=nλ+∑i=1∞1−Gθxi;ξ−nexpλ+∑i=1∞expλ1−Gθxi;ξ1−Gθxi;ξ−nexpλexp1−eλ1−exp1−eλ,∂ℓ∂θ=nθ+∑i=1∞logGxi;ξ−λ∑i=1∞Gθxi;ξlogGxi;ξ−λ∑i=1∞expλ1−Gθxi;ξGθxi;ξlogGxi;ξ,∂ℓ∂ξ=∑i=1∞giξgxi;ξ+θ−1∑i=1∞GiξGxi;ξ−λθ∑i=1∞Gθ−1xi;ξGiξ−λθ∑i=1∞expλ1−Gθxi;ξGθ−1xi;ξGiξ,where giξ=∂/∂ξgxi;ξ and Giξ=∂/∂ξGxi;ξ are derivatives of column vectors of the same dimension of ξ, and by setting ϕλ=0,ϕθ=0, and ϕξ=0, the MLEs can be yielded by solving the above equations simultaneously.Proposition 7.
A randomly selected sample of sizenis under EBell-G; then, the score vectorϕλ,ϕθ,ϕξis given by


## 4. Layout of the EBellE Model

Due to the closed form solution of many real problems and simplicity, exponential distribution is commonly employed in lifetime testing as well as reliability analysis. However, the exponential distribution failed to yield better fits when hazard rates are nonconstant. However, several studies showed that extended exponential distribution or when it is used as baseline model provides better fits [22–24]. In the present study, we used exponential distribution as a baseline model which yielded flexibility in both pdf and hrf shapes given in Figures 2 and 3, respectively. We now define the EBellE distribution by taking the exponential model as baseline, with the following expression of densities gx=αexp−αx and Gx=1−exp−αx for x>0 and α>0, by setting these densities in (7) and (8) yielded the following expression for the proposed EBellE distribution. Then, the cdf and pdf are of the EBellE distribution, respectively.(59)Fx=1−exp−eλ1−e−λ1−e−αxθ1−exp1−eλ,(60)fx=λθαe−αx1−e−αxθ−1expλ1−1−e−αxθ×exp−eλ1−e−λ1−e−αxθ1−exp1−eλ−1.Figure 2
Plots of EBellE density for some parametric values.Figure 3
Plots of hazard rate of EBellE for some parametric values.Proposition 8.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, its cdf is given by in Eq. (7):Proposition 9.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, its pdf is given by in Eq. (8):The exponential distribution quantile function becomesQu=QGz=−1/αlog1−z; using (9), z=1−λ−1loglog1−u1−exp1−eλ+expλθ−1. The quantile function of x can be expressed as(61)Qu=−1αlog1−1−λ−1loglog1−u1−exp1−eλ+expλθ−1.The sf and the hrf of the EBellE model can be obtained as(62)sx=exp−eλ1−e−λ1−e−αxθ−exp1−eλ1−exp1−eλ,hx=λθαe−αx1−e−αxθ−1expλ1−1−e−αxθexp−eλ1−e−λ1−e−αxθexp−eλ1−e−λ1−e−αxθ−exp1−eλ.


### 4.1. Properties of the EBellE Model

First, we will deduce linear representation of EBellE density to obtain useful properties of that model. By using Eq. (12),(63)fx=∑v=0∞wvαθv+1exp−αx1−exp−αxθv+1−1,1−exp−αxθv+1−1=∑p=0∞−1pθv+1−1pexp−αpx.After applying Eq. (10), it reduces to(64)fx=∑p=0∞wpπx;αp+1,where πx;αp+1 is a exp-exponential density with αp+1 parameter and(65)wp=∑v=0∞wvθv+1p+1−1pθv+1−1p.It is obvious from Eq. (25) that the EBellE density is a linear combination of exponential densities, and therefore, one can obtain several properties using Eq. (25).


#### 4.1.1. The Expression ofrth Moment

(66)μr′=∑p=0∞wpΓr+1αp+1−r.Proposition 10.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itsrth moment can be written as by taking into account Eq. (25):By settingr  = 1 yielded the mean of the EBellE model.


#### 4.1.2. The Expression ofsth Incomplete Moment

(67)φst=∑p=0∞wpγs+1,αp+1tαp+1−s.Proposition 11.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itssth incomplete moment can be written as by taking into account Eq. (25):By settings  = 1 yielded the first incomplete moment of the EBellE model. Table 2 shows the first four raw moments, central moments, coefficient of variation, coefficient of kurtosis, and Pearson’s coefficient of skewness for some parametric values. Six different scenarios of parametric values are used in order to compute different measures of dispersion. S-1 = α=2.5,θ=1.0,λ=0.2, S-2 = α=1.5,θ=1.4,λ=1.2, S-3 =  α=0.85,θ=0.75,λ=1.2, S-4 = α=0.85,θ=2.5,λ=0.2, S-5 = α=4.85,θ=0.22,λ=0.12, and S−6=α=2.5,θ=3.85,λ=1. The following relationship is used to obtain the central moments: μ2=μ2′−μ1′2,  μ3=μ3′−3μ1′μ2′+2μ1′3, and μ4=μ4′−4μ3′μ1′+6μ2′μ1′2−3μ1′4. The moment-based measure of skewness and kurtosis is obtained by using β1=μ32/μ23 and β2=μ4/μ22, respectively. Pearson’s coefficient of skewness is simply square root of β1, and coefficient of kurtosis is computed as β2−3. Furthermore, we present the mean, variance, skewness, and kurtosis of EBellE in Figures 4 and 5, respectively, utilizing these results. Some plots of Bonferroni and Lorenz curve are also depicted in Figure 6.Table 2
Measures of dispersion of the EBellE model for some parametric value.
MeasuresS-1S-2S-3S-4S-5S-6μ1′0.35910.34010.30611.83070.05860.4655μ2′0.27220.27400.40645.01980.01910.3953μ3′0.31810.41591.139618.82370.01090.4156μ4′0.50220.97244.875490.86620.00870.5970μ20.14320.15830.31271.66820.01570.1786μ30.11740.21500.82373.52580.00790.0654μ40.20610.55673.682220.26730.00650.1962β14.692211.639622.18262.677916.27710.7506β210.04622.201637.64987.283126.32596.1551CS2.16623.41174.709801.63644.03450.8663CK7.046319.201634.64984.283123.32593.1551Figure 4
Graphical illustration of mean (a) and variance (b) of the EBellE model.
(a)(b)Figure 5
Graphical illustration of skewness (a) and kurtosis (b) of the EBellE model.
(a)(b)Figure 6
Plot of f Bonferroni (a) and Lorenz (b) curves of EBellE for some parametric values.
(a)(b)


#### 4.1.3. The Expression ofr th Conditional Moment

From actuarial prospective, conditional moments are important; let EBellE beλ,θ,α for x>0 and λ,θ,α>0; then, its r th conditional moment can be written by using Equation (64):(68)Exr|X>t=1F¯EBellEt∫t∞xrfxdx,Exr|X>t=1F¯EBellE∑p=0∞wpγr+1,αp+1tαp+1−r.


#### 4.1.4. Two Expression of MGF

LetX∼ EBellE λ,θ,α for x>0 and λ,θ,α>0; then, its moment generating function by using Wright generalization hypergeometric function is given as(69)Ψqp=α1,A1,…………,αp,Apβ1,B1,…………,βq,Bq;x=∑n=0∞Πj=1pΓαj+AjnΠj=1qΓβj+Bjnxnn!,(70)Mt=ap+1∑p=0∞wp∫0∞exptxexp−ap+1xdx.ConsiderI=∫0∞exptxexp−ap+1xdx and exptx=∑m=0∞tm/m!xm; equation (70) is reduced to(71)I=1ap+1m∑m=0∞Γm+1tmm!.By using (70), Equation (71) yielded as(72)I=1ap+1mΨ011,1−;t,Mt=∑p=0∞wp1ap+1mΨ011,1−;t.The other representation of mgf is given by(73)Mt=ap+1∑p=0∞wpap+1−t−1.


#### 4.1.5. Order Statistics

Thesth moment of order statistic can be obtained by using (41):(74)EXi:ns=∑j=0∞Qi:njθj+1∫0∞xsgxGθj+1−1xdx.Simplification yielded the expression ofsth moments of order statistics:(75)EXi:ns=∑p=0∞tpΓs+1αp+1−s,where tp=∑j=0∞−1pθj+1−1pQi:njθj+1/p+1.To study the distributional behaviour of the set of observation, we can use minimum and maximum (min-max) plot of the order statistics. Min-max plot depends on extreme order statistics, and it is introduced to capture all information not only about the tails of the distribution but also about the whole distribution of the data. Figure7 shows the min and the max order statistics for some parametric values and depends on EX1:n and EXn:n, respectively.Figure 7
Min-Max plot of order statistics of the EBellE model for some parametric values.


#### 4.1.6. Stochastic Ordering

LetX and Y be the two rvs from EBellE distribution with the assumption previously illustrated in Section 3 given that λ1<λ2, and for X1≤lrX2, f1x/f2x shall be decreasing in x if and the only if the following results holds:(76)ddxlogf1xf2x=θαexp−αx1−exp−αxθ−1×λ2−λ1+λ2eλ21−1−exp−αxθ−λ1eλ11−1−exp−αxθ<0.


#### 4.1.7. Rényi Entropy

The Rényi entropy for the EBellE model by using Eq. (22) given under and δ>0 and δ≠1:(77)IδX=1−δ−1log∑p=0∞Qpδ+p−1,where Qp=αδ−1∑b=0∞Qb−1pθb+δθ−1p, and the graphical demonstration of Rényi entropy of EBellE at varying of the parameters is given in Figure 8.Figure 8
Plot of Rényi entropy of the EBellE model for some parametric values.


#### 4.1.8. Reliability

Reliability is an important measure, and several applications are documented in the field of economics, physical science, and engineering. Reliability enables us to determine the failure probability at certain point in a time. LetX1 and X2 be the two random variable following the EBellE distribution. The component fails if the applied stress exceeds its strength; if X1>X2, the component will perform satisfactory. Reliability is defined by the following expression. Here, we derive the reliability of the EBellE model when X1 and X2 have independent f1x;λ1,θ,α and F2x;λ2,θ,α with identical scale α and shape θ parameters. The reliability is given by(78)R=∫0∞F2xf1xdx.By using equations (14) and (15), we get the pdf and the cdf of EBellE, respectively, as follows:(79)f1x;λ1,θ,α=∑v=0∞wvλ1αθv+1exp−αx1−exp−αxθv+1−1,F2x;λ1,θ,α=∑t=0∞wtλ21−exp−αxθt+1.Hence,(80)Iα,β,θ,v,t=∫0∞αexp−αx1−exp−αxθv+1−1×1−exp−αxθt+1dx.Therefore,(81)Iα,β,θ,v,t=α∑z=0∞sz∫0∞exp−α1+zxdx.By using gamma function, the above expression is reduced to(82)Iα,β,θ,v,t=∑z=0∞sz1+z−1,where sz=−1zθv+t+2−1z.


### 4.2. Estimation

The log-likelihood function for the vector of parametersϕ=λ,θ,α⊤ for model given in (60) is given by(83)ℓϕ=nlogλθα−α∑i=1∞xi+θ−1∑i=1∞log1−e−αxi+λ∑i=1∞1−1−e−αxiθ−nexpλ+∑i=1∞expλ1−1−e−αxiθ−nlog1−exp1−eλ.The components of the score vectorUϕ are(84)Uλ=nλ+∑i=1∞1−1−e−αxiθ−nexpλ+∑i=1∞expλ1−1−e−αxiθ1−1−e−αxiθ−nexpλexp1−eλ1−exp1−eλ,Uθ=nθ+∑i=1∞log1−e−αxi−λ∑i=1∞1−e−αxiθlog1−e−αxi−λ∑i=1∞expλ1−1−e−αxiθ1−e−αxiθlog1−e−αxi,Uα=nα−∑i=1∞xi+θ−1∑i=1∞xiexp−αxi1−e−αxi−λθ∑i=1∞xiexp−αxi1−e−αxiθ−1−λθ∑i=1∞xiexp−αxiexpλ1−1−e−αxiθ.By settingUλ=0,Uθ=0 and Uα=0, the MLEs can be yielded by solving the above equations simultaneously.


## 4.1. Properties of the EBellE Model

First, we will deduce linear representation of EBellE density to obtain useful properties of that model. By using Eq. (12),(63)fx=∑v=0∞wvαθv+1exp−αx1−exp−αxθv+1−1,1−exp−αxθv+1−1=∑p=0∞−1pθv+1−1pexp−αpx.After applying Eq. (10), it reduces to(64)fx=∑p=0∞wpπx;αp+1,where πx;αp+1 is a exp-exponential density with αp+1 parameter and(65)wp=∑v=0∞wvθv+1p+1−1pθv+1−1p.It is obvious from Eq. (25) that the EBellE density is a linear combination of exponential densities, and therefore, one can obtain several properties using Eq. (25).


### 4.1.1. The Expression ofrth Moment

(66)μr′=∑p=0∞wpΓr+1αp+1−r.Proposition 10.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itsrth moment can be written as by taking into account Eq. (25):By settingr  = 1 yielded the mean of the EBellE model.


### 4.1.2. The Expression ofsth Incomplete Moment

(67)φst=∑p=0∞wpγs+1,αp+1tαp+1−s.Proposition 11.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itssth incomplete moment can be written as by taking into account Eq. (25):By settings  = 1 yielded the first incomplete moment of the EBellE model. Table 2 shows the first four raw moments, central moments, coefficient of variation, coefficient of kurtosis, and Pearson’s coefficient of skewness for some parametric values. Six different scenarios of parametric values are used in order to compute different measures of dispersion. S-1 = α=2.5,θ=1.0,λ=0.2, S-2 = α=1.5,θ=1.4,λ=1.2, S-3 =  α=0.85,θ=0.75,λ=1.2, S-4 = α=0.85,θ=2.5,λ=0.2, S-5 = α=4.85,θ=0.22,λ=0.12, and S−6=α=2.5,θ=3.85,λ=1. The following relationship is used to obtain the central moments: μ2=μ2′−μ1′2,  μ3=μ3′−3μ1′μ2′+2μ1′3, and μ4=μ4′−4μ3′μ1′+6μ2′μ1′2−3μ1′4. The moment-based measure of skewness and kurtosis is obtained by using β1=μ32/μ23 and β2=μ4/μ22, respectively. Pearson’s coefficient of skewness is simply square root of β1, and coefficient of kurtosis is computed as β2−3. Furthermore, we present the mean, variance, skewness, and kurtosis of EBellE in Figures 4 and 5, respectively, utilizing these results. Some plots of Bonferroni and Lorenz curve are also depicted in Figure 6.Table 2
Measures of dispersion of the EBellE model for some parametric value.
MeasuresS-1S-2S-3S-4S-5S-6μ1′0.35910.34010.30611.83070.05860.4655μ2′0.27220.27400.40645.01980.01910.3953μ3′0.31810.41591.139618.82370.01090.4156μ4′0.50220.97244.875490.86620.00870.5970μ20.14320.15830.31271.66820.01570.1786μ30.11740.21500.82373.52580.00790.0654μ40.20610.55673.682220.26730.00650.1962β14.692211.639622.18262.677916.27710.7506β210.04622.201637.64987.283126.32596.1551CS2.16623.41174.709801.63644.03450.8663CK7.046319.201634.64984.283123.32593.1551Figure 4
Graphical illustration of mean (a) and variance (b) of the EBellE model.
(a)(b)Figure 5
Graphical illustration of skewness (a) and kurtosis (b) of the EBellE model.
(a)(b)Figure 6
Plot of f Bonferroni (a) and Lorenz (b) curves of EBellE for some parametric values.
(a)(b)


### 4.1.3. The Expression ofr th Conditional Moment

From actuarial prospective, conditional moments are important; let EBellE beλ,θ,α for x>0 and λ,θ,α>0; then, its r th conditional moment can be written by using Equation (64):(68)Exr|X>t=1F¯EBellEt∫t∞xrfxdx,Exr|X>t=1F¯EBellE∑p=0∞wpγr+1,αp+1tαp+1−r.


### 4.1.4. Two Expression of MGF

LetX∼ EBellE λ,θ,α for x>0 and λ,θ,α>0; then, its moment generating function by using Wright generalization hypergeometric function is given as(69)Ψqp=α1,A1,…………,αp,Apβ1,B1,…………,βq,Bq;x=∑n=0∞Πj=1pΓαj+AjnΠj=1qΓβj+Bjnxnn!,(70)Mt=ap+1∑p=0∞wp∫0∞exptxexp−ap+1xdx.ConsiderI=∫0∞exptxexp−ap+1xdx and exptx=∑m=0∞tm/m!xm; equation (70) is reduced to(71)I=1ap+1m∑m=0∞Γm+1tmm!.By using (70), Equation (71) yielded as(72)I=1ap+1mΨ011,1−;t,Mt=∑p=0∞wp1ap+1mΨ011,1−;t.The other representation of mgf is given by(73)Mt=ap+1∑p=0∞wpap+1−t−1.


### 4.1.5. Order Statistics

Thesth moment of order statistic can be obtained by using (41):(74)EXi:ns=∑j=0∞Qi:njθj+1∫0∞xsgxGθj+1−1xdx.Simplification yielded the expression ofsth moments of order statistics:(75)EXi:ns=∑p=0∞tpΓs+1αp+1−s,where tp=∑j=0∞−1pθj+1−1pQi:njθj+1/p+1.To study the distributional behaviour of the set of observation, we can use minimum and maximum (min-max) plot of the order statistics. Min-max plot depends on extreme order statistics, and it is introduced to capture all information not only about the tails of the distribution but also about the whole distribution of the data. Figure7 shows the min and the max order statistics for some parametric values and depends on EX1:n and EXn:n, respectively.Figure 7
Min-Max plot of order statistics of the EBellE model for some parametric values.


### 4.1.6. Stochastic Ordering

LetX and Y be the two rvs from EBellE distribution with the assumption previously illustrated in Section 3 given that λ1<λ2, and for X1≤lrX2, f1x/f2x shall be decreasing in x if and the only if the following results holds:(76)ddxlogf1xf2x=θαexp−αx1−exp−αxθ−1×λ2−λ1+λ2eλ21−1−exp−αxθ−λ1eλ11−1−exp−αxθ<0.


### 4.1.7. Rényi Entropy

The Rényi entropy for the EBellE model by using Eq. (22) given under and δ>0 and δ≠1:(77)IδX=1−δ−1log∑p=0∞Qpδ+p−1,where Qp=αδ−1∑b=0∞Qb−1pθb+δθ−1p, and the graphical demonstration of Rényi entropy of EBellE at varying of the parameters is given in Figure 8.Figure 8
Plot of Rényi entropy of the EBellE model for some parametric values.


### 4.1.8. Reliability

Reliability is an important measure, and several applications are documented in the field of economics, physical science, and engineering. Reliability enables us to determine the failure probability at certain point in a time. LetX1 and X2 be the two random variable following the EBellE distribution. The component fails if the applied stress exceeds its strength; if X1>X2, the component will perform satisfactory. Reliability is defined by the following expression. Here, we derive the reliability of the EBellE model when X1 and X2 have independent f1x;λ1,θ,α and F2x;λ2,θ,α with identical scale α and shape θ parameters. The reliability is given by(78)R=∫0∞F2xf1xdx.By using equations (14) and (15), we get the pdf and the cdf of EBellE, respectively, as follows:(79)f1x;λ1,θ,α=∑v=0∞wvλ1αθv+1exp−αx1−exp−αxθv+1−1,F2x;λ1,θ,α=∑t=0∞wtλ21−exp−αxθt+1.Hence,(80)Iα,β,θ,v,t=∫0∞αexp−αx1−exp−αxθv+1−1×1−exp−αxθt+1dx.Therefore,(81)Iα,β,θ,v,t=α∑z=0∞sz∫0∞exp−α1+zxdx.By using gamma function, the above expression is reduced to(82)Iα,β,θ,v,t=∑z=0∞sz1+z−1,where sz=−1zθv+t+2−1z.


## 4.1.1. The Expression ofrth Moment

(66)μr′=∑p=0∞wpΓr+1αp+1−r.Proposition 10.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itsrth moment can be written as by taking into account Eq. (25):By settingr  = 1 yielded the mean of the EBellE model.


## 4.1.2. The Expression ofsth Incomplete Moment

(67)φst=∑p=0∞wpγs+1,αp+1tαp+1−s.Proposition 11.
LetX∼EBellEλ,θ,α, forx>0andλ,θ,α>0; then, itssth incomplete moment can be written as by taking into account Eq. (25):By settings  = 1 yielded the first incomplete moment of the EBellE model. Table 2 shows the first four raw moments, central moments, coefficient of variation, coefficient of kurtosis, and Pearson’s coefficient of skewness for some parametric values. Six different scenarios of parametric values are used in order to compute different measures of dispersion. S-1 = α=2.5,θ=1.0,λ=0.2, S-2 = α=1.5,θ=1.4,λ=1.2, S-3 =  α=0.85,θ=0.75,λ=1.2, S-4 = α=0.85,θ=2.5,λ=0.2, S-5 = α=4.85,θ=0.22,λ=0.12, and S−6=α=2.5,θ=3.85,λ=1. The following relationship is used to obtain the central moments: μ2=μ2′−μ1′2,  μ3=μ3′−3μ1′μ2′+2μ1′3, and μ4=μ4′−4μ3′μ1′+6μ2′μ1′2−3μ1′4. The moment-based measure of skewness and kurtosis is obtained by using β1=μ32/μ23 and β2=μ4/μ22, respectively. Pearson’s coefficient of skewness is simply square root of β1, and coefficient of kurtosis is computed as β2−3. Furthermore, we present the mean, variance, skewness, and kurtosis of EBellE in Figures 4 and 5, respectively, utilizing these results. Some plots of Bonferroni and Lorenz curve are also depicted in Figure 6.Table 2
Measures of dispersion of the EBellE model for some parametric value.
MeasuresS-1S-2S-3S-4S-5S-6μ1′0.35910.34010.30611.83070.05860.4655μ2′0.27220.27400.40645.01980.01910.3953μ3′0.31810.41591.139618.82370.01090.4156μ4′0.50220.97244.875490.86620.00870.5970μ20.14320.15830.31271.66820.01570.1786μ30.11740.21500.82373.52580.00790.0654μ40.20610.55673.682220.26730.00650.1962β14.692211.639622.18262.677916.27710.7506β210.04622.201637.64987.283126.32596.1551CS2.16623.41174.709801.63644.03450.8663CK7.046319.201634.64984.283123.32593.1551Figure 4
Graphical illustration of mean (a) and variance (b) of the EBellE model.
(a)(b)Figure 5
Graphical illustration of skewness (a) and kurtosis (b) of the EBellE model.
(a)(b)Figure 6
Plot of f Bonferroni (a) and Lorenz (b) curves of EBellE for some parametric values.
(a)(b)


## 4.1.3. The Expression ofr th Conditional Moment

From actuarial prospective, conditional moments are important; let EBellE beλ,θ,α for x>0 and λ,θ,α>0; then, its r th conditional moment can be written by using Equation (64):(68)Exr|X>t=1F¯EBellEt∫t∞xrfxdx,Exr|X>t=1F¯EBellE∑p=0∞wpγr+1,αp+1tαp+1−r.


## 4.1.4. Two Expression of MGF

LetX∼ EBellE λ,θ,α for x>0 and λ,θ,α>0; then, its moment generating function by using Wright generalization hypergeometric function is given as(69)Ψqp=α1,A1,…………,αp,Apβ1,B1,…………,βq,Bq;x=∑n=0∞Πj=1pΓαj+AjnΠj=1qΓβj+Bjnxnn!,(70)Mt=ap+1∑p=0∞wp∫0∞exptxexp−ap+1xdx.ConsiderI=∫0∞exptxexp−ap+1xdx and exptx=∑m=0∞tm/m!xm; equation (70) is reduced to(71)I=1ap+1m∑m=0∞Γm+1tmm!.By using (70), Equation (71) yielded as(72)I=1ap+1mΨ011,1−;t,Mt=∑p=0∞wp1ap+1mΨ011,1−;t.The other representation of mgf is given by(73)Mt=ap+1∑p=0∞wpap+1−t−1.


## 4.1.5. Order Statistics

Thesth moment of order statistic can be obtained by using (41):(74)EXi:ns=∑j=0∞Qi:njθj+1∫0∞xsgxGθj+1−1xdx.Simplification yielded the expression ofsth moments of order statistics:(75)EXi:ns=∑p=0∞tpΓs+1αp+1−s,where tp=∑j=0∞−1pθj+1−1pQi:njθj+1/p+1.To study the distributional behaviour of the set of observation, we can use minimum and maximum (min-max) plot of the order statistics. Min-max plot depends on extreme order statistics, and it is introduced to capture all information not only about the tails of the distribution but also about the whole distribution of the data. Figure7 shows the min and the max order statistics for some parametric values and depends on EX1:n and EXn:n, respectively.Figure 7
Min-Max plot of order statistics of the EBellE model for some parametric values.


## 4.1.6. Stochastic Ordering

LetX and Y be the two rvs from EBellE distribution with the assumption previously illustrated in Section 3 given that λ1<λ2, and for X1≤lrX2, f1x/f2x shall be decreasing in x if and the only if the following results holds:(76)ddxlogf1xf2x=θαexp−αx1−exp−αxθ−1×λ2−λ1+λ2eλ21−1−exp−αxθ−λ1eλ11−1−exp−αxθ<0.


## 4.1.7. Rényi Entropy

The Rényi entropy for the EBellE model by using Eq. (22) given under and δ>0 and δ≠1:(77)IδX=1−δ−1log∑p=0∞Qpδ+p−1,where Qp=αδ−1∑b=0∞Qb−1pθb+δθ−1p, and the graphical demonstration of Rényi entropy of EBellE at varying of the parameters is given in Figure 8.Figure 8
Plot of Rényi entropy of the EBellE model for some parametric values.


## 4.1.8. Reliability

Reliability is an important measure, and several applications are documented in the field of economics, physical science, and engineering. Reliability enables us to determine the failure probability at certain point in a time. LetX1 and X2 be the two random variable following the EBellE distribution. The component fails if the applied stress exceeds its strength; if X1>X2, the component will perform satisfactory. Reliability is defined by the following expression. Here, we derive the reliability of the EBellE model when X1 and X2 have independent f1x;λ1,θ,α and F2x;λ2,θ,α with identical scale α and shape θ parameters. The reliability is given by(78)R=∫0∞F2xf1xdx.By using equations (14) and (15), we get the pdf and the cdf of EBellE, respectively, as follows:(79)f1x;λ1,θ,α=∑v=0∞wvλ1αθv+1exp−αx1−exp−αxθv+1−1,F2x;λ1,θ,α=∑t=0∞wtλ21−exp−αxθt+1.Hence,(80)Iα,β,θ,v,t=∫0∞αexp−αx1−exp−αxθv+1−1×1−exp−αxθt+1dx.Therefore,(81)Iα,β,θ,v,t=α∑z=0∞sz∫0∞exp−α1+zxdx.By using gamma function, the above expression is reduced to(82)Iα,β,θ,v,t=∑z=0∞sz1+z−1,where sz=−1zθv+t+2−1z.


## 4.2. Estimation

The log-likelihood function for the vector of parametersϕ=λ,θ,α⊤ for model given in (60) is given by(83)ℓϕ=nlogλθα−α∑i=1∞xi+θ−1∑i=1∞log1−e−αxi+λ∑i=1∞1−1−e−αxiθ−nexpλ+∑i=1∞expλ1−1−e−αxiθ−nlog1−exp1−eλ.The components of the score vectorUϕ are(84)Uλ=nλ+∑i=1∞1−1−e−αxiθ−nexpλ+∑i=1∞expλ1−1−e−αxiθ1−1−e−αxiθ−nexpλexp1−eλ1−exp1−eλ,Uθ=nθ+∑i=1∞log1−e−αxi−λ∑i=1∞1−e−αxiθlog1−e−αxi−λ∑i=1∞expλ1−1−e−αxiθ1−e−αxiθlog1−e−αxi,Uα=nα−∑i=1∞xi+θ−1∑i=1∞xiexp−αxi1−e−αxi−λθ∑i=1∞xiexp−αxi1−e−αxiθ−1−λθ∑i=1∞xiexp−αxiexpλ1−1−e−αxiθ.By settingUλ=0,Uθ=0 and Uα=0, the MLEs can be yielded by solving the above equations simultaneously.


## 5. Actuarial Measures

### 5.1. Value at Risk

Value-at-risk or quantile risk or simplyVaR is the extensively used as a standard finial market risk measure. It plays an important role in many business decisions; the uncertainty regarding foreign market, commodity price, and government policies can affect significantly firm earnings. The loss portfolio value is specified by the certain degree of confidence say q90%,95%or99%. VaR of random variable X is simply the qth quantile of its cdf. If X follows the EBellE model; then, its VaR is defined by the following expression:(85)Qu=−1αlog1−1−λ−1loglog1−q1−exp1−eλ+expλθ−1.


### 5.2. Expected Shortfall

The other important financial risk measure is called an expected-shortfall (ES) introduced in [25] and generally considered a better measure than value-at-risk. It is defined by the following expression:(86)ESqx=1q∫0qVaRxdx.For0<q<1, using Eq. (85) in Eq. (86), yielded ES for EBellE. In Figure 9, the graphical representation of VaR and ES measures for some parameter combinations is presented.Figure 9
Plot of VaR (a) and ES (b) of EBellE distribution for some parametric values.
(a)(b)


### 5.3. Tail Value at Risk

The problem of risk measurement is one of the most important problems in the risk management. From finance and insurance prospective, Tail value-at-risk (TVaR) or tail conditional expectation or conditional tail expectation is an important measure and define as the expected value of the loss, given the loss is greater than the VaR:(87)TVaRqx=11−q∫VaRq∞xfxdx.By using (25) in (35) yielded tail value-at-risk as(88)TVaRqx=1−q−1∑p=0∞wpγ2,αp+1VaRqαp+1−1.


### 5.4. Tail Variance

Tail variance (TV) has yet another important risk measure because it considers the variability of the risk along the tail of distribution; it is defined as from the following expression:(89)TVqx=EX2|X>xq−TVaRq2.ConsiderI=EX2|X>xq:(90)I=TVaRqx=11−q∫VaRq∞x2fxdx,I=TVaRqx=1−q−1∑p=0∞wpγ3,αp+1VaRqαp+1−2.Using (88) and (90) in (89), we obtain the expression for tail variance for the EBellW model.


### 5.5. Tail Variance Premium

The TVP is the mixture of both central tendency as well as dispersion statistics. It is defined by the following:(91)TVPqX=TVaRq+δTVq,where 0<δ<1. Using expressions (89) and (88) in (91), we obtain the tail variance premium for the EBellW model.


#### 5.5.1. Numerical Illustration of VaR and ES

Here, we demonstrate the numerical as well as graphical presentation of the two important risk measures ES and VaR. The comparative study of ES and VaR of the proposed EBellE model with their counterpart exponentiated exponential Poisson (EEP) and exponentiated exponential (EE) model is performed by taking MLEs’ estimates of the parameters for the models in all datasets. It is worth emphasising that a model with higher values of the risk measures is said to have a heavier tail. Table3 provides the numerical illustration of the ES and VaR for EBellE and EEP and EE model of all three datasets and yielded that the EBellE model has higher values of both the risk measures as compared to their counterpart EEP and EE model. The graphical demonstration of the models from Figures 10–12 also revealed that the proposed model has slightly heavier tail than EEP and EE model. The reader should refer to Chan et al. [26] for detail discussion of VaR and ES and their computation by using an R-Programming Language. A sample of 100 is randomly drawn, and the effect of shape and scale parameters of the proposed models is underlined for both risk measures. Various combinations of the scale and shape parameters are executed I = α=2.1,θ=1.1,λ=0.22, II = α=1.4,θ=0.5,λ=1.2, III = α=0.6,θ=0.5,λ=0.5, and IV = α=0.4,θ=2,λ=2.5 and change in the curve of VaR and ES are illustrated in Figure 4.Table 3
The detailed summary of ES and VaR of the datasets.
qData 1Data 2Data 3EBellEEEPEEEBellEEEPEEEBellEEEPEEES0.5053.61452.00950.251131.69997.948132.3503.72193.33822.70680.6054.99153.38451.710133.96399.956134.1064.15453.74293.08690.6556.38554.77653.193136.192101.971135.8844.62434.18163.49980.7057.81656.20454.716138.409104.019137.7065.14114.66283.95170.7559.30857.69156.301140.640106.130139.5965.71925.19824.45080.8060.89559.26857.977142.919108.345141.5896.38035.80565.00980.8562.62960.98459.786145.287110.725143.7347.16166.51475.64840.9064.60862.92761.801147.816113.379146.1168.13597.38246.40180.9567.05865.29964.182150.656116.561148.9229.49098.54967.34850.9969.96368.03366.726153.517120.178151.90911.419810.10868.4328VaR0.5068.73467.10866.235156.895120.079151.6188.30357.62486.72470.6071.58169.95569.327160.872124.043155.2739.55248.78947.83480.6574.69973.06372.686165.034128.320159.23411.012610.14069.10540.7078.20876.54576.416169.485133.053163.62212.765511.745710.58510.7582.29780.57580.673174.376138.458168.61914.944513.713312.34900.8087.28985.45085.714179.950144.904174.52617.789916.235114.52350.8593.81991.74892.016186.649153.099181.89621.793819.694217.34540.90103.403100.823100.649195.463164.722191.97428.197425.028121.34720.95121.320117.303115.012209.589185.624199.43641.690835.646528.22750.99171.119161.023147.586249.567243.769235.64689.187369.551344.2822Figure 10
Plot of ES (a) and VaR (b) of EBellE distribution Data-1.
(a)(b)Figure 11
Plot of ES (a) and VaR (b) of EBellE distribution Data-2.
(a)(b)Figure 12
Plot of ES (a) and VaR (b) of EBellE distribution Data-3.
(a)(b)


## 5.1. Value at Risk

Value-at-risk or quantile risk or simplyVaR is the extensively used as a standard finial market risk measure. It plays an important role in many business decisions; the uncertainty regarding foreign market, commodity price, and government policies can affect significantly firm earnings. The loss portfolio value is specified by the certain degree of confidence say q90%,95%or99%. VaR of random variable X is simply the qth quantile of its cdf. If X follows the EBellE model; then, its VaR is defined by the following expression:(85)Qu=−1αlog1−1−λ−1loglog1−q1−exp1−eλ+expλθ−1.


## 5.2. Expected Shortfall

The other important financial risk measure is called an expected-shortfall (ES) introduced in [25] and generally considered a better measure than value-at-risk. It is defined by the following expression:(86)ESqx=1q∫0qVaRxdx.For0<q<1, using Eq. (85) in Eq. (86), yielded ES for EBellE. In Figure 9, the graphical representation of VaR and ES measures for some parameter combinations is presented.Figure 9
Plot of VaR (a) and ES (b) of EBellE distribution for some parametric values.
(a)(b)


## 5.3. Tail Value at Risk

The problem of risk measurement is one of the most important problems in the risk management. From finance and insurance prospective, Tail value-at-risk (TVaR) or tail conditional expectation or conditional tail expectation is an important measure and define as the expected value of the loss, given the loss is greater than the VaR:(87)TVaRqx=11−q∫VaRq∞xfxdx.By using (25) in (35) yielded tail value-at-risk as(88)TVaRqx=1−q−1∑p=0∞wpγ2,αp+1VaRqαp+1−1.


## 5.4. Tail Variance

Tail variance (TV) has yet another important risk measure because it considers the variability of the risk along the tail of distribution; it is defined as from the following expression:(89)TVqx=EX2|X>xq−TVaRq2.ConsiderI=EX2|X>xq:(90)I=TVaRqx=11−q∫VaRq∞x2fxdx,I=TVaRqx=1−q−1∑p=0∞wpγ3,αp+1VaRqαp+1−2.Using (88) and (90) in (89), we obtain the expression for tail variance for the EBellW model.


## 5.5. Tail Variance Premium

The TVP is the mixture of both central tendency as well as dispersion statistics. It is defined by the following:(91)TVPqX=TVaRq+δTVq,where 0<δ<1. Using expressions (89) and (88) in (91), we obtain the tail variance premium for the EBellW model.


### 5.5.1. Numerical Illustration of VaR and ES

Here, we demonstrate the numerical as well as graphical presentation of the two important risk measures ES and VaR. The comparative study of ES and VaR of the proposed EBellE model with their counterpart exponentiated exponential Poisson (EEP) and exponentiated exponential (EE) model is performed by taking MLEs’ estimates of the parameters for the models in all datasets. It is worth emphasising that a model with higher values of the risk measures is said to have a heavier tail. Table3 provides the numerical illustration of the ES and VaR for EBellE and EEP and EE model of all three datasets and yielded that the EBellE model has higher values of both the risk measures as compared to their counterpart EEP and EE model. The graphical demonstration of the models from Figures 10–12 also revealed that the proposed model has slightly heavier tail than EEP and EE model. The reader should refer to Chan et al. [26] for detail discussion of VaR and ES and their computation by using an R-Programming Language. A sample of 100 is randomly drawn, and the effect of shape and scale parameters of the proposed models is underlined for both risk measures. Various combinations of the scale and shape parameters are executed I = α=2.1,θ=1.1,λ=0.22, II = α=1.4,θ=0.5,λ=1.2, III = α=0.6,θ=0.5,λ=0.5, and IV = α=0.4,θ=2,λ=2.5 and change in the curve of VaR and ES are illustrated in Figure 4.Table 3
The detailed summary of ES and VaR of the datasets.
qData 1Data 2Data 3EBellEEEPEEEBellEEEPEEEBellEEEPEEES0.5053.61452.00950.251131.69997.948132.3503.72193.33822.70680.6054.99153.38451.710133.96399.956134.1064.15453.74293.08690.6556.38554.77653.193136.192101.971135.8844.62434.18163.49980.7057.81656.20454.716138.409104.019137.7065.14114.66283.95170.7559.30857.69156.301140.640106.130139.5965.71925.19824.45080.8060.89559.26857.977142.919108.345141.5896.38035.80565.00980.8562.62960.98459.786145.287110.725143.7347.16166.51475.64840.9064.60862.92761.801147.816113.379146.1168.13597.38246.40180.9567.05865.29964.182150.656116.561148.9229.49098.54967.34850.9969.96368.03366.726153.517120.178151.90911.419810.10868.4328VaR0.5068.73467.10866.235156.895120.079151.6188.30357.62486.72470.6071.58169.95569.327160.872124.043155.2739.55248.78947.83480.6574.69973.06372.686165.034128.320159.23411.012610.14069.10540.7078.20876.54576.416169.485133.053163.62212.765511.745710.58510.7582.29780.57580.673174.376138.458168.61914.944513.713312.34900.8087.28985.45085.714179.950144.904174.52617.789916.235114.52350.8593.81991.74892.016186.649153.099181.89621.793819.694217.34540.90103.403100.823100.649195.463164.722191.97428.197425.028121.34720.95121.320117.303115.012209.589185.624199.43641.690835.646528.22750.99171.119161.023147.586249.567243.769235.64689.187369.551344.2822Figure 10
Plot of ES (a) and VaR (b) of EBellE distribution Data-1.
(a)(b)Figure 11
Plot of ES (a) and VaR (b) of EBellE distribution Data-2.
(a)(b)Figure 12
Plot of ES (a) and VaR (b) of EBellE distribution Data-3.
(a)(b)


## 5.5.1. Numerical Illustration of VaR and ES

Here, we demonstrate the numerical as well as graphical presentation of the two important risk measures ES and VaR. The comparative study of ES and VaR of the proposed EBellE model with their counterpart exponentiated exponential Poisson (EEP) and exponentiated exponential (EE) model is performed by taking MLEs’ estimates of the parameters for the models in all datasets. It is worth emphasising that a model with higher values of the risk measures is said to have a heavier tail. Table3 provides the numerical illustration of the ES and VaR for EBellE and EEP and EE model of all three datasets and yielded that the EBellE model has higher values of both the risk measures as compared to their counterpart EEP and EE model. The graphical demonstration of the models from Figures 10–12 also revealed that the proposed model has slightly heavier tail than EEP and EE model. The reader should refer to Chan et al. [26] for detail discussion of VaR and ES and their computation by using an R-Programming Language. A sample of 100 is randomly drawn, and the effect of shape and scale parameters of the proposed models is underlined for both risk measures. Various combinations of the scale and shape parameters are executed I = α=2.1,θ=1.1,λ=0.22, II = α=1.4,θ=0.5,λ=1.2, III = α=0.6,θ=0.5,λ=0.5, and IV = α=0.4,θ=2,λ=2.5 and change in the curve of VaR and ES are illustrated in Figure 4.Table 3
The detailed summary of ES and VaR of the datasets.
qData 1Data 2Data 3EBellEEEPEEEBellEEEPEEEBellEEEPEEES0.5053.61452.00950.251131.69997.948132.3503.72193.33822.70680.6054.99153.38451.710133.96399.956134.1064.15453.74293.08690.6556.38554.77653.193136.192101.971135.8844.62434.18163.49980.7057.81656.20454.716138.409104.019137.7065.14114.66283.95170.7559.30857.69156.301140.640106.130139.5965.71925.19824.45080.8060.89559.26857.977142.919108.345141.5896.38035.80565.00980.8562.62960.98459.786145.287110.725143.7347.16166.51475.64840.9064.60862.92761.801147.816113.379146.1168.13597.38246.40180.9567.05865.29964.182150.656116.561148.9229.49098.54967.34850.9969.96368.03366.726153.517120.178151.90911.419810.10868.4328VaR0.5068.73467.10866.235156.895120.079151.6188.30357.62486.72470.6071.58169.95569.327160.872124.043155.2739.55248.78947.83480.6574.69973.06372.686165.034128.320159.23411.012610.14069.10540.7078.20876.54576.416169.485133.053163.62212.765511.745710.58510.7582.29780.57580.673174.376138.458168.61914.944513.713312.34900.8087.28985.45085.714179.950144.904174.52617.789916.235114.52350.8593.81991.74892.016186.649153.099181.89621.793819.694217.34540.90103.403100.823100.649195.463164.722191.97428.197425.028121.34720.95121.320117.303115.012209.589185.624199.43641.690835.646528.22750.99171.119161.023147.586249.567243.769235.64689.187369.551344.2822Figure 10
Plot of ES (a) and VaR (b) of EBellE distribution Data-1.
(a)(b)Figure 11
Plot of ES (a) and VaR (b) of EBellE distribution Data-2.
(a)(b)Figure 12
Plot of ES (a) and VaR (b) of EBellE distribution Data-3.
(a)(b)


## 6. Designing of GASP under the EBellE Model

Saving time and cost is attributed to the sampling method. Certain quality control checks are implemented either accepting or rejecting a lot under various sampling plans. This section based on the illustration of GASP under the assumption when the lifetime distribution of an item followed a EBellE model with known parameterλ and θ having cdf in Eq. (96). In a GASP, let a random sample of size n be taken and distributed in such a way; that is, n=r×g and r items for each group are kept on life testing under predefined time. If the number of failures in each group are higher than the acceptance number c, the performed experiment is truncated. The reader is referred to the work of Aslam et al. [27] and Khan and Alqarni [28] for simple illustration of GASP and application to real data. Designing the GASP reduced both the time and cost. Several lifetime traditional and extended models are used [27, 29–32] in designing the GASP by taking into account the quality parameter as mean or median; usually, for skewed distribution, median is preferable [27].The GASP is simply the extension of ordinary sampling plan (OSP), i.e., the GASP tends to OSP by replacingr=1, and thus, n=g [33].GASP is based on the following form; first of all, selectg and allocate predefine r items to each group so that the sample of size of the lot will be n=r×g. Secondly, select c and t0 representing the acceptance number and the experiment time, respectively. Thirdly, do experiment simultaneously for g groups and record the number of failure for each group. Finally, conclusion is drawn either accepting or rejecting the lot; the lot is accepted if there is no more than c failure occurring in each and every group; otherwise, reject a lot. The accepting probability of a lot yielded by the following expression:(92)pap=∑i=0cripi1−pr−ig,where the probability that an item in a group fail before t0 is denoted by p and yielded by inserting (61) in (96). Let the lifetime of an item or product follow a EBellE with known parameters θ and λ, with cdf given for t>0:(93)Ft=1−exp−eλ1−e−λ1−e−αtθ1−exp1−eλ.qf of the EBellE model using (61) is given by, and if p=0.5 yielded median lifetime distribution for a product or item,(94)m=−1αlog1−1−λ−1loglog1−u1−exp1−eλ+expλ1/θ,By takingη as follows,(95)η=log1−1−λ−1loglog1−u1−exp1−eλ+expλ1/θ.Eq. (94) becomes by replacing η; henceforth, α=−η/m and t=m0a1, m=−1/αη. The ratio of a of product mean lifetime ti and the specified life time m/m0 can be used to express the quality level of product. By replacing α=−η/m and t=m0a1 in Eq. (96) yielded the probability of failure given by(96)p=1−exp−eλ1−e−λ1−e−αtθ1−exp1−eλ.From Eq. (96), for chosen θ and λ, p can be determined when a1 and r2 are specified, where r2=m/m0. Here, we define the two failure probabilities say p1 and p2 corresponding to the consumer risk and producer risk, respectively. For a given specific values of the parameters θ and λ, r2, a1, β, and γ, we need to evaluate the value of c and g that satisfy the following two equation simultaneously:(97)pap1|m/m0=r1=∑i=0crip1i1−p1r−ig≤β,pap2|m/m0=r2=∑i=0crip2i1−p2r−ig≥1−γ,where the mean ratio at consumer’s risk and at producer’s risk, respectively, is denoted by r1 and r2 and the probability of failure to be used in the above expression is as follows:(98)p1=1−exp−eλ1−e−λ1−ea1ηθ1−exp1−eλ,p2=1−exp−eλ1−e−λ1−ea1ηr2−1θ1−exp1−eλ.From Tables4 and 5, with β=0.25, a1  = 0.5, and r2  = 4 and taking r  = 5, there are 40 groups or 200 40×5=200; total units are needed for lifetime testing. While on the contrary, significant reduction can be observed in groups or number of units to be tested under the identical circumstances when r  = 10; a total of 3 groups or 30 3×10=30 item are needed for life testing. Here, we prefer the group size 10. Similarly, when β  = 0.25, a1  = 1, and r2  = 4 and taking r  = 5, there are 7 groups or 35 7×5=35; total units are needed for life testing. While, on the contrary, in the number of units to be tested under the identical circumstances when r  = 10, a total of 2 groups or 20 2×10=20, items are needed for life testing. Here, we prefer the group size 10.Table 4
GASP under the EBellE model,θ=1 and λ=1.25, showing minimum g and c.
βr2r=5r=10a1=0.5a1=1a1=0.5a1=1gcp(a)gcp(a)gcp(a)gcp(a)0.252––––––––––––44030.9860730.9721330.9701240.96246820.9815320.9559220.9565130.97488820.9917320.979220.9792130.98990.12––––––––––––46630.97701230.9527530.9507550.984361220.97241230.9881530.9875230.950381220.9875420.9721320.9689230.98000.052––––––––––––48530.97059540.98451740.9798750.978161620.96331530.9852730.9825230.950381620.9834520.9652320.9689230.98000.012––––––––––––413130.954914640.97632640.96931050.9688613130.98992330.97741030.9751540.980382420.9752720.95171030.9910330.9701Remark: a large sample size in required cells contains hyphens (–).Table 5
GASP under the EBellE model,θ=1 and λ=1.50, showing minimum g and c.
βr2r=5r=10a1=0.5a1=1a1=0.5a1=1gcp(a)gcp(a)gcp(a)gcp(a)0.252––––––––––––44130.9866730.9735330.9718240.96446820.9824320.9578220.9584130.97618820.9921320.98220.9802130.99050.12––––––––––––46830.97791230.955530.9535550.985461320.97151230.9888530.9883230.952881320.9872420.9734320.9705230.98110.052––––––––––––48830.97159540.98551740.9814750.979661620.96511530.9861730.9836230.952881620.9843520.9669320.9705230.98110.012––––––––––––413530.956514640.97782640.97161050.9709613530.99032330.97871030.9767540.981682520.9755720.954520.9513330.9718Remark: a large sample size in required cells contains hyphens (–).


## 7. Simulation Analysis

Simulation analysis is very important tools in statistics and used to determine the performance of estimates over predefine replication at varying sample sizes. So, this section is primarily based on simulation analysis in order to underline the performance parameter estimates of the proposed EBellE model. A simulation process is replicated 1000 times with at varying sample sizes,n  = 25, 50, 100, and 500. In Table 6, various combinations of the parameter α, θ, and λ are considered, say scenario I = α=1.5,θ=0.3,λ=0.5, scenario II =  α=0.15,θ=0.5,λ=0.5, and scenario III =  α=1,θ=0.2,λ=0.17. The finding of the simulation analysis yielded that bias, mean square error (MSE), and average width (AW) of the confidence interval of the parameters reduced as sample size increases. On the contrary, the coverage probabilities (CPs) touch 95% nominal level. So, therefore, the MLEs and their asymptotic results can be used for estimating and constructing confidence intervals for proposed EBellE model parameters. Readers are referred to the work of Sigal et al. [34] for simple but comprehensive way in designing Monte Carlo simulation study by using R-programming language:(99)MSEΘ^=∑r=11,000Θi^−Θ21,000,BiasΘ^=∑r=11,000Θi^1,000−Θ.Table 6
Biases, MSEs, CPs, and AWs for different scenarios.
Scenario IScenario IIScenario IIInαθλnαθλnαθλBiases250.2820.0500.152250.0060.0630.176250.1620.0670.375500.0980.0280.105500.0020.0380.112500.0380.0480.2891000.0460.0150.0501000.0100.0170.042100−0.0070.0310.1965000.0280.0050.0095000.0010.000-0.0095000.0080.0070.043MSE251.3980.0100.162250.0070.0230.198250.6690.0090.258500.5840.0050.122500.0040.0120.145500.2040.0050.1691000.2620.0020.0731000.0010.0060.0941000.0010.0060.0945000.0550.0010.0215000.0000.0010.0295000.0000.0010.029CPs250.9680.9550.982250.9720.9630.959250.9570.9380.930500.9650.9600.980500.9760.9570.971500.9250.9270.9041000.9550.9530.9691000.9400.9540.9721000.9280.9270.9035000.9300.9450.9505000.9400.9480.9645000.8810.9650.945AWs255.0010.3962.461250.4050.6272.721253.1580.3162.375503.1230.2761.782500.2790.4552.047501.8730.2221.7511002.0780.1961.2761000.0820.3271.4851001.1700.1541.2715000.8870.0880.5735000.0820.1490.6715000.4430.0670.607


## 8. Practical Implementation of the Proposed EBellE Model

### 8.1. Actuarial Data

Here, we demonstrate the flexibility and usefulness of the proposed EBellE model by practical means. Three insurance claim datasets are used; the first two datasets based on unemployment claims from July 2008 to April 2013, reported by the Department of Labour, Licencing, and Regulation, USA. The dataset consists of 21 variables; we used the variable 5 that is new claims filed and variable 12 with total observation for each variable is 58. The dataset was also used by [15]. The third data deal with upheld most frequent complaints such as nonrenewal of insurance, and no fault claims commonly against vehicle insurance company over two-year period as a proportion of their overall business. The dataset was also used by Khan et al. [21]. The descriptive summary of all three datasets is shown in Table 7 and consists of sample size n, minimum claim x0, maximum claim xn, lower Q1 and upper Q3, quartile deviations, mean x¯, median x˜, standard deviation σ, measures of skewness Sk, and kurtosis K. Total time on test (TTT) plots of the datasets is illustrated in Figure 13, revealing that the first two datasets have increasing hazard rate function, while the third dataset has decreasing (increasing) hazard rate function.Table 7
Descriptive summary of datasets.
nx0Q1x˜x¯σQ3xnSkKData 15829.00053.25063.5070.67032.64574.750222.002.43610.622Data 258102.00133.00153.00155.3031.899176.00267.000.6084.0694Data 3891.04802.61607.094014.07925.26615.374204.175.31237.969Figure 13
TTT Plots of EBellE of data 1–4.The comparative study is carried out with several modified well-established exponential models, namely, exponentiated exponential Poisson (EEP) [35], alpha power exponentiated exponential (APEE) [15], Transmuted generalized exponential (TGE) [36], gamma exponentiated exponential (GEE) [37], exponential (E), exponentiated exponential (EE), Marshal Olkin exponential (MOE) [38], exponentiated Weibull (EW) [39], odd Weibull exponential (OWE) [19], Weibull (W), Kumaraswamy exponential (KE) [40], beta exponential (BE) [41], Tope Leone exponential (TLE) [42], and Nadarajah Haghigh (NH) [43] distributions.All statistical computational work is carried out usingR-programming language. Table 8 shows the MLEs and standard errors (S.E) of the estimates of the fitted models of the data sets. Table 9 demonstrated the commonly used well-known model selection information criterion, namely, AIC, CAIC, BIC, and HQIC with important measures including Anderson–Darling A∗, Cramér–von Mises W∗, and Kolmogrov–Smirnov (K–S) test and p value of all three datasets. The analysis of the datasets revealed the proposed three-parameter EBellE model, outperforming compared to several well established models. A model having higher p values and least information criterion and A∗ and W∗, and the K-S value is considered as best models among all other comparative models. TTT plots of the respective datasets are shown in Figure 13. Likewise, plots of the estimated pdf, cdf, hrf, and sf for the four datasets are provided in Figures 14–17. Additionally, PP-plots are presented in Figure 18.Table 8
Estimated parameters and S.Es of insurance data.
Dist.ParameterData 1Data 2Data 3EstimatesS.EEstimatesS.EEstimatesS.EEBellEα^0.0280.0070.00970.0020.0220.010θ^10.863.44213.673.3921.1740.124λ^1.340.3252.3340.3451.5100.274EEPα^0.0320.0070.0150.0020.0260.011β^11.653.79418.014.3571.0870.119λ^3.1831.4104.3261.4564.4001.496APEEα^0.0320.0070.0200.0070.0260.010β^0.0400.0570.0080.0240.0120.018γ^11.483.71442.1132.121.0870.119TGEα^0.0410.0060.0280.0070.0440.010β^0.6200.2620.9220.3830.7720.169λ^13.634.44580.0965.400.9380.117GEEα^0.0510.0130.1320.0040.0550.010β^11.978.1362.8190.4661.5020.738λ^1.3541.13919.490.8750.5570.277EEα^0.0480.0060.0430.0170.0630.010β^16.085.250405.2903.60.8370.117MOEα^0.0660.0080.0350.0030.0280.011a^72.1341.02160.957.750.2130.106OWEα^0.0030.0010.1070.0680.0030.001a^14.287.2800.0110.00511.112.873b^1.9110.1800.2450.1560.7630.055Wα^0.5850.0810.6420.0830.1270.028β^40.3319.1224.499.4510.8230.061KEα^0.0730.0230.1270.0030.0080.005a^34.9327.330.0810.0420.8190.074b^0.5430.2470.0490.0076.1333.409BEα^0.0760.0250.0200.0030.0150.017a^33.0024.5834.7089.9000.8180.107b^0.5150.2352.1460.6853.8714.198TLEα^0.0240.0030.0110.0010.0310.005a^15.985.19017.354.2160.8380.117NHα^0.0030.0010.0030.0010.2290.070β^3.9000.8742.0220.3470.5490.076Eα^0.0140.0020.0060.0010.0710.008Table 9
The statisticsℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and p value for datasets.
Dist.−2ℓ^AICCAICBICHQICA∗W∗K.SP valueData 1EBellE264.79535.57536.01541.75537.980.6030.1080.0960.655EEP265.31536.62537.06542.80539.030.7010.1270.0990.619APEE265.31536.62537.06542.80539.030.7030.1280.1000.610TGE266.36538.71539.16544.89541.120.8930.1640.1020.576GEE267.65541.31541.75547.49543.711.1320.2090.1180.395EE267.49538.97539.19543.09540.581.0900.2010.1130.448MOE274.32552.64552.85556.76554.242.2180.4000.1400.204OWE572.33569.93570.37576.11572.333.3120.6080.1870.034W291.24586.47586.69590.59588.082.0650.3800.3320.000KE266.73539.46539.90545.64541.860.9910.1810.1150.430BE266.65539.30539.74545.48541.700.9750.1780.1130.448TLE267.49538.97539.19543.09540.581.0910.2020.1130.445NH295.63595.27595.48599.39596.872.7810.5110.3710.000E304.97611.93612.01613.99612.741.7550.3240.3870.000Data 2EBellE281.25568.51568.95574.69570.910.18870.01840.05230.9973EEP284.62575.24575.68581.42577.640.31780.03680.10890.4967APEE281.86569.71570.15575.89572.120.40170.04970.07950.8565TGE281.56569.12569.57575.30571.530.38410.04640.08400.8078GEE281.33568.66569.11574.84571.070.21410.01880.05450.9954EE283.74571.49571.71575.61573.090.70220.09530.12610.3146MOE291.12586.24586.46590.36587.850.20460.01970.16870.0737OWE294.13594.27594.71600.45596.681.38430.18750.14810.1571W291.24586.47586.69590.59588.082.06540.37990.33200.0001KE357.54721.08721.52727.26723.490.20360.01700.52910.0000BE282.53571.05571.50577.24573.460.28800.03090.08270.8220TLE293.44590.88591.10595.00592.490.29130.03140.17560.0560NH341.65687.30687.51691.42688.900.19820.01610.49920.0001E350.62703.23703.30705.29704.030.20910.01820.48180.0001Data 3EBellE313.42632.85633.13640.31635.861.2650.1850.1130.188EEP314.86635.73636.01643.19638.741.4100.2050.1140.182APEE314.86635.73636.01643.19638.741.4100.2040.1140.182TGE318.91643.81644.09651.28646.821.8020.2620.1180.156GEE322.69651.39651.67658.85654.392.1910.3230.1210.134EE323.55651.10651.24656.08653.112.2710.3350.1250.114MOE316.42636.83636.97641.81638.841.1560.1610.1270.104OWE322.55651.10651.38658.57654.111.9790.2860.1380.061W320.41644.82644.96649.80646.831.8610.2690.1260.110KE321.66649.32649.60656.79652.332.0300.2960.1180.152BE323.12652.24652.53659.71655.252.2260.3280.1170.162TLE323.55651.10651.24656.08653.112.2710.3350.1250.113NH315.90635.80635.94640.78637.811.2860.1820.1220.132E324.38650.76650.80653.25651.762.2320.3290.1620.017Figure 14
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 1.Figure 15
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 2.Figure 16
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 3.Figure 17
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 4.Figure 18
P-P plots of EBellE of data 1–4.


### 8.2. GASP

Recently, Almarashi et al. [29] designed a GASP under Marshall–Olkin–Kumaraswamy exponential distribution by using the data of breaking strength of carbon fibers. The data consist the 50 observed values with mean (1.975) and median (1.190) breaking strength of carbon fibers, respectively. Under the K–S test, the maximum distance between actual and fitted yielded as 0.0681 with p value 0.9743 under Marshall–Olkin–Kumaraswamy exponential distribution. We used the same data as data-4, and our proposed three-parameter EBellE model is slightly better fit compared to four-parameter Marshall–Olkin–Kumaraswamy exponential distribution [29] as K-S test as 0.0680 with improved p value as 0.9749. The estimated parameters (SEs), namely, α^  = 0.3913 (0.1308), θ^  = 0.9088 (0.2114), and λ^  = 0.3431 (0.5766). Table 10 shows the GAPS under the EBellE model at MLEs’ values showing minimum g and c when r  = 5 and r  = 10, with a1  = 0.5 and 1. The analysis of the data yielded from Table 10, with β=0.25, a1  = 1, and r2  = 4 and taking r  = 5, there are 7 groups or 35 7×5=35; total units are needed for lifetime testing. While, on the contrary, significant reduction can be observed in groups or number of units to be tested under the identical circumstances when r  = 10; a total of 2 groups or 20 2×10=20 item are needed for life testing. Here, we prefer the group size as 10. When the true median life increases, the number of groups decreases and the operating characteristics values increases under the EBellE model.Table 10
GASP under the EBellE model,θ^=0.9088 and λ^=0.3432, showing minimum g and c Data 4.
βr2r=5r=10a1=0.5a1=1a1=0.5a1=1gcp(a)gcp(a)gcp(a)gcp(a)0.252––––––––––––43830.9808730.9668330.9585240.95476720.9766730.9908330.9880130.96788720.9887320.9726220.9686130.98600.12––––––––––––46330.96837340.98501340.9763550.980061220.96021230.9842530.9800340.983681220.9806420.9636320.9533230.97210.052––––––––––––48230.95909540.98061640.9709750.972261520.95051530.9804730.9722440.978281520.9758520.9547320.9533230.97210.012––––––––––––4–––14640.97032540.95491050.9605612530.98402330.97001030.9605540.972882320.96322330.98831030.9842330.9585Remark: a large sample size in required cells contains hyphens (–).


### 8.3. Concluding remarks

We introduced and documented the new flexible family of distributions called exponentiated Bell-G family. We also derived general mathematical properties of the proposed family, namely, linear representation of the density, random variable generation, reliability properties, ordinary moments, generating function, probability weighted moment, entropies, order statistics, reverse order statistics, entropies measures, upper records values, stochastic ordering, and estimation of parameters. We also illustrated the important actuarial measures and design of GASP. We also implemented the new proposed generator to the four real datasets by taking exponential distribution as a special case. The analysis of the data yielded that the new generator is found to be superior compared to their counterparts. [44].


## 8.1. Actuarial Data

Here, we demonstrate the flexibility and usefulness of the proposed EBellE model by practical means. Three insurance claim datasets are used; the first two datasets based on unemployment claims from July 2008 to April 2013, reported by the Department of Labour, Licencing, and Regulation, USA. The dataset consists of 21 variables; we used the variable 5 that is new claims filed and variable 12 with total observation for each variable is 58. The dataset was also used by [15]. The third data deal with upheld most frequent complaints such as nonrenewal of insurance, and no fault claims commonly against vehicle insurance company over two-year period as a proportion of their overall business. The dataset was also used by Khan et al. [21]. The descriptive summary of all three datasets is shown in Table 7 and consists of sample size n, minimum claim x0, maximum claim xn, lower Q1 and upper Q3, quartile deviations, mean x¯, median x˜, standard deviation σ, measures of skewness Sk, and kurtosis K. Total time on test (TTT) plots of the datasets is illustrated in Figure 13, revealing that the first two datasets have increasing hazard rate function, while the third dataset has decreasing (increasing) hazard rate function.Table 7
Descriptive summary of datasets.
nx0Q1x˜x¯σQ3xnSkKData 15829.00053.25063.5070.67032.64574.750222.002.43610.622Data 258102.00133.00153.00155.3031.899176.00267.000.6084.0694Data 3891.04802.61607.094014.07925.26615.374204.175.31237.969Figure 13
TTT Plots of EBellE of data 1–4.The comparative study is carried out with several modified well-established exponential models, namely, exponentiated exponential Poisson (EEP) [35], alpha power exponentiated exponential (APEE) [15], Transmuted generalized exponential (TGE) [36], gamma exponentiated exponential (GEE) [37], exponential (E), exponentiated exponential (EE), Marshal Olkin exponential (MOE) [38], exponentiated Weibull (EW) [39], odd Weibull exponential (OWE) [19], Weibull (W), Kumaraswamy exponential (KE) [40], beta exponential (BE) [41], Tope Leone exponential (TLE) [42], and Nadarajah Haghigh (NH) [43] distributions.All statistical computational work is carried out usingR-programming language. Table 8 shows the MLEs and standard errors (S.E) of the estimates of the fitted models of the data sets. Table 9 demonstrated the commonly used well-known model selection information criterion, namely, AIC, CAIC, BIC, and HQIC with important measures including Anderson–Darling A∗, Cramér–von Mises W∗, and Kolmogrov–Smirnov (K–S) test and p value of all three datasets. The analysis of the datasets revealed the proposed three-parameter EBellE model, outperforming compared to several well established models. A model having higher p values and least information criterion and A∗ and W∗, and the K-S value is considered as best models among all other comparative models. TTT plots of the respective datasets are shown in Figure 13. Likewise, plots of the estimated pdf, cdf, hrf, and sf for the four datasets are provided in Figures 14–17. Additionally, PP-plots are presented in Figure 18.Table 8
Estimated parameters and S.Es of insurance data.
Dist.ParameterData 1Data 2Data 3EstimatesS.EEstimatesS.EEstimatesS.EEBellEα^0.0280.0070.00970.0020.0220.010θ^10.863.44213.673.3921.1740.124λ^1.340.3252.3340.3451.5100.274EEPα^0.0320.0070.0150.0020.0260.011β^11.653.79418.014.3571.0870.119λ^3.1831.4104.3261.4564.4001.496APEEα^0.0320.0070.0200.0070.0260.010β^0.0400.0570.0080.0240.0120.018γ^11.483.71442.1132.121.0870.119TGEα^0.0410.0060.0280.0070.0440.010β^0.6200.2620.9220.3830.7720.169λ^13.634.44580.0965.400.9380.117GEEα^0.0510.0130.1320.0040.0550.010β^11.978.1362.8190.4661.5020.738λ^1.3541.13919.490.8750.5570.277EEα^0.0480.0060.0430.0170.0630.010β^16.085.250405.2903.60.8370.117MOEα^0.0660.0080.0350.0030.0280.011a^72.1341.02160.957.750.2130.106OWEα^0.0030.0010.1070.0680.0030.001a^14.287.2800.0110.00511.112.873b^1.9110.1800.2450.1560.7630.055Wα^0.5850.0810.6420.0830.1270.028β^40.3319.1224.499.4510.8230.061KEα^0.0730.0230.1270.0030.0080.005a^34.9327.330.0810.0420.8190.074b^0.5430.2470.0490.0076.1333.409BEα^0.0760.0250.0200.0030.0150.017a^33.0024.5834.7089.9000.8180.107b^0.5150.2352.1460.6853.8714.198TLEα^0.0240.0030.0110.0010.0310.005a^15.985.19017.354.2160.8380.117NHα^0.0030.0010.0030.0010.2290.070β^3.9000.8742.0220.3470.5490.076Eα^0.0140.0020.0060.0010.0710.008Table 9
The statisticsℓ^, AIC, CAIC, BIC, HQIC, A∗, W∗, K-S, and p value for datasets.
Dist.−2ℓ^AICCAICBICHQICA∗W∗K.SP valueData 1EBellE264.79535.57536.01541.75537.980.6030.1080.0960.655EEP265.31536.62537.06542.80539.030.7010.1270.0990.619APEE265.31536.62537.06542.80539.030.7030.1280.1000.610TGE266.36538.71539.16544.89541.120.8930.1640.1020.576GEE267.65541.31541.75547.49543.711.1320.2090.1180.395EE267.49538.97539.19543.09540.581.0900.2010.1130.448MOE274.32552.64552.85556.76554.242.2180.4000.1400.204OWE572.33569.93570.37576.11572.333.3120.6080.1870.034W291.24586.47586.69590.59588.082.0650.3800.3320.000KE266.73539.46539.90545.64541.860.9910.1810.1150.430BE266.65539.30539.74545.48541.700.9750.1780.1130.448TLE267.49538.97539.19543.09540.581.0910.2020.1130.445NH295.63595.27595.48599.39596.872.7810.5110.3710.000E304.97611.93612.01613.99612.741.7550.3240.3870.000Data 2EBellE281.25568.51568.95574.69570.910.18870.01840.05230.9973EEP284.62575.24575.68581.42577.640.31780.03680.10890.4967APEE281.86569.71570.15575.89572.120.40170.04970.07950.8565TGE281.56569.12569.57575.30571.530.38410.04640.08400.8078GEE281.33568.66569.11574.84571.070.21410.01880.05450.9954EE283.74571.49571.71575.61573.090.70220.09530.12610.3146MOE291.12586.24586.46590.36587.850.20460.01970.16870.0737OWE294.13594.27594.71600.45596.681.38430.18750.14810.1571W291.24586.47586.69590.59588.082.06540.37990.33200.0001KE357.54721.08721.52727.26723.490.20360.01700.52910.0000BE282.53571.05571.50577.24573.460.28800.03090.08270.8220TLE293.44590.88591.10595.00592.490.29130.03140.17560.0560NH341.65687.30687.51691.42688.900.19820.01610.49920.0001E350.62703.23703.30705.29704.030.20910.01820.48180.0001Data 3EBellE313.42632.85633.13640.31635.861.2650.1850.1130.188EEP314.86635.73636.01643.19638.741.4100.2050.1140.182APEE314.86635.73636.01643.19638.741.4100.2040.1140.182TGE318.91643.81644.09651.28646.821.8020.2620.1180.156GEE322.69651.39651.67658.85654.392.1910.3230.1210.134EE323.55651.10651.24656.08653.112.2710.3350.1250.114MOE316.42636.83636.97641.81638.841.1560.1610.1270.104OWE322.55651.10651.38658.57654.111.9790.2860.1380.061W320.41644.82644.96649.80646.831.8610.2690.1260.110KE321.66649.32649.60656.79652.332.0300.2960.1180.152BE323.12652.24652.53659.71655.252.2260.3280.1170.162TLE323.55651.10651.24656.08653.112.2710.3350.1250.113NH315.90635.80635.94640.78637.811.2860.1820.1220.132E324.38650.76650.80653.25651.762.2320.3290.1620.017Figure 14
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 1.Figure 15
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 2.Figure 16
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 3.Figure 17
Plots of estimated density, estimated cdf, estimated hrf, and failure rate for data 4.Figure 18
P-P plots of EBellE of data 1–4.


## 8.2. GASP

Recently, Almarashi et al. [29] designed a GASP under Marshall–Olkin–Kumaraswamy exponential distribution by using the data of breaking strength of carbon fibers. The data consist the 50 observed values with mean (1.975) and median (1.190) breaking strength of carbon fibers, respectively. Under the K–S test, the maximum distance between actual and fitted yielded as 0.0681 with p value 0.9743 under Marshall–Olkin–Kumaraswamy exponential distribution. We used the same data as data-4, and our proposed three-parameter EBellE model is slightly better fit compared to four-parameter Marshall–Olkin–Kumaraswamy exponential distribution [29] as K-S test as 0.0680 with improved p value as 0.9749. The estimated parameters (SEs), namely, α^  = 0.3913 (0.1308), θ^  = 0.9088 (0.2114), and λ^  = 0.3431 (0.5766). Table 10 shows the GAPS under the EBellE model at MLEs’ values showing minimum g and c when r  = 5 and r  = 10, with a1  = 0.5 and 1. The analysis of the data yielded from Table 10, with β=0.25, a1  = 1, and r2  = 4 and taking r  = 5, there are 7 groups or 35 7×5=35; total units are needed for lifetime testing. While, on the contrary, significant reduction can be observed in groups or number of units to be tested under the identical circumstances when r  = 10; a total of 2 groups or 20 2×10=20 item are needed for life testing. Here, we prefer the group size as 10. When the true median life increases, the number of groups decreases and the operating characteristics values increases under the EBellE model.Table 10
GASP under the EBellE model,θ^=0.9088 and λ^=0.3432, showing minimum g and c Data 4.
βr2r=5r=10a1=0.5a1=1a1=0.5a1=1gcp(a)gcp(a)gcp(a)gcp(a)0.252––––––––––––43830.9808730.9668330.9585240.95476720.9766730.9908330.9880130.96788720.9887320.9726220.9686130.98600.12––––––––––––46330.96837340.98501340.9763550.980061220.96021230.9842530.9800340.983681220.9806420.9636320.9533230.97210.052––––––––––––48230.95909540.98061640.9709750.972261520.95051530.9804730.9722440.978281520.9758520.9547320.9533230.97210.012––––––––––––4–––14640.97032540.95491050.9605612530.98402330.97001030.9605540.972882320.96322330.98831030.9842330.9585Remark: a large sample size in required cells contains hyphens (–).


## 8.3. Concluding remarks

We introduced and documented the new flexible family of distributions called exponentiated Bell-G family. We also derived general mathematical properties of the proposed family, namely, linear representation of the density, random variable generation, reliability properties, ordinary moments, generating function, probability weighted moment, entropies, order statistics, reverse order statistics, entropies measures, upper records values, stochastic ordering, and estimation of parameters. We also illustrated the important actuarial measures and design of GASP. We also implemented the new proposed generator to the four real datasets by taking exponential distribution as a special case. The analysis of the data yielded that the new generator is found to be superior compared to their counterparts. [44].



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*Source: 2489998-2022-06-09.xml*