# The Novel Bivariate Distribution: Statistical Properties and Real Data Applications

Authors: B. I. Mohammed; Abdulaziz S. Alghamdi; Hassan M. Aljohani; Md. Moyazzem Hossain
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This article proposes a novel class of bivariate distributions that are completely defined by stating their conditionals as Poisson exponential distributions. Numerous statistical properties of this distribution are also examined here, including the conditional probability mass function (PMF) and moments of the new class. The techniques of maximum likelihood and pseudolikelihood are used to estimate the model parameters. Additionally, the effectiveness of the bivariate Poisson exponential conditional (BPEC) distribution is compared to that of the bivariate Poisson conditional (BPC), the bivariate Poisson (BP), the bivariate Poisson–Lindley (BPL), and the bivariate negative binomial (BNB) distributions using a real-world dataset. The findings of Akaike information criterion (AIC) and Bayesian information criterion (BIC) reveal that the BPEC distribution performs better than the other distributions considered in this study. As a result, the authors claim that this distribution may be used to fit dependent and overspread count data.


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## Body

## 1. Introduction

In many areas of application, it is appropriate to study discrete bivariate variables. For example, problems arise in many social, economic, and physical phenomena [1], and in insurance risk applications, those number of cases in distinctive classifications will be regularly randomized (the readers are referred to Wu and Yuen [2], Yuen et al. [3], and Morata [4] for more details). Several authors have discussed these problems from different points of view, which include traffic accidents by Cacoullos and Papageorgiou [5] and Papageorgiou [6, 7] and the problem associated with crime utilizing the method of Miethe et al. [8]. Also, Lee [9] and Karlis and Ntzoufras [10] modeled scores “points and goals” of two competing teams in sports and pointed out that they are highly correlated. Modeling dependence on goals scored by teams competing in international football matches was studied by McHale and Scarf [11], and evaluated risks and spot errors using scarce data were discussed by Ahooyi et al. [12]. Several discrete bivariate models have been proposed in the literature (see, for example, Marshall and Olkin [13], Mishra [14], Özel [15–17], Reilly and Sapkota [18], Lee and Cha [19], and Jiang et al. [20]). The specific conditional distributions are one of the most important ways to get flexible bivariate distributions. Moreover, the important role of functional equations has been emphasized in establishing results in this regard which is highlighted by Castillo and Galambos [21–23], Arnold [24], Arnold et al. [25–27], Kottas et al. [28], and Gharib and Mohammed [29]. The use of this type of distribution in risk analysis and economics is relatively new; however, some applications were done by Sarabia et al. [30, 31].In this paper, another class of bivariate model for Poisson exponential conditionals will be considered. A discrete random variableX is said to have a one-parameter Poisson exponential distribution (PED) for modeling countable data if its probability mass function (PMF) is(1)PZ=z=α1+αz,α>0,z=0,1,….If the parameter of the Poisson model follows a continuous exponential distribution, then equation (1) is a mixture of Poisson and exponential distributions denoted by Z∼PEα. This distribution is applicable to biological datasets, traffic datasets, thunderstorm datasets, and other discrete datasets. The scientific properties and estimation for parameter have been examined by Fazal and Bashir [32]; also, its requisition turns out that it will be a great substitution cost from claiming Poisson and Lindley distributions. In this paper, a new class of bivariate distribution has been proposed which is fully characterized by specifying its conditionals as Poisson exponential distribution. Finally, the performance of this distribution is compared with other distributions considering a real-life dataset.


## 2. Bivariate Poisson Exponential Conditionals

Consider a general bivariate modelJ,K whose conditional distributions must satisfy the following two conditions:(2)J|K=k∼PEη1k,(3)K|J=j∼PEη2j,where η1k and η2j are some positive functions and PE denotes a Poisson exponential distribution. These equations lead us to discuss the next theorem.Theorem 1.
The discrete bivariate model withJ|K=k∼PEη1k and K|J=j∼PEη2j can be described by the following distribution:(4)PJ,Kj,k=Nθ1,θ2,θ3−1expθ1k−θ3jk+θ2j,j,k=0,1,…,θ1,θ2<0,θ3∈ℛ,where Nθ1,θ2,θ3−1 is the normalizing constant such that PJ,Kj,k summates to 1.Proof.
According to (2) and (3), we can write the joint density P(j, k) as a product of a marginal and a conditional density in both ways to get(5)η1k1+η1kjhKk=η2j1+η2jkhJj,where hJj and hKk are the marginal PMFs of J and K, respectively.
Denoting(6)gk=logη1khKk,(7)fj=logη2jhJj,equation (5) readily reduces to(8)gk−xlog1+η1k−fj−klog1+η2j=0,which is a special case of the functional equation ∑m=1nfmxgmy=0, whose most general solution is given by Aczel [33] as follows:(9)α2j=exp−θ1+θ3j−1,…,η1k=exp−θ2+θ3k−1.
Substituting these expressions in (6) and (7), we can get the marginal PMF as(10)hJj=Nθ1,θ2,θ3−1exp−θ1+θ3j−1exp−θ2j,j=0,1,…,(11)hKk=Nθ1,θ2,θ3−1exp−θ2+θ3k−1exp−θ1k,k=0,1,….
Finally, in accordance with (10) and (11), the class of discrete bivariate distribution with Poisson exponential conditionals is that given by (4), which describes the complete class of the BPEC distribution that has the three parameters θ1,θ2 (intensity parameters for K and J, respectively), and θ3∈ℛ (interaction or dependence parameter), where θ3=0 corresponds to independence between J and K.
Figure1 shows the three-dimensional curve of the BPLC given by (4) for the special cases for θ1,θ2,andθ3.Figure 1
The three-dimensional curve of the BPLC under different scenarios. (a) BPEC withθ1=−0.2,θ2=−0.7,andθ3=0.3. (b) BPEC with θ1=−0.2,θ2=−0.7,andθ3=−0.3. (c) BPEC with θ1=−1.7,θ2=−1.5,andθ3=0.4. (d) BPEC with θ1=−1.7,θ2=−1.5,andθ3=−0.4.
(a)(b)(c)(d)


## 3. Properties of the Bivariate Poisson Exponential Class

In this part, the fundamental properties of the new bivariate distribution are contemplated.We first know that the class (4) has the three parameters θ1,θ2, and θ3, while Nθ1,θ2,θ3−1 is the normalizing constant and is given by(12)Nθ1,θ2,θ3=∑j=0∞∑k=0∞Expθ1k−θ3jk+θ2j=∑j=0∞e−θ1+θ3−θ2jeθ1+θ3j−1.


### 3.1. Conditional PMF and Moments

The particular manifestations of the conditional distributions to the new model would provide(13)PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,(14)PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(15)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexp−θ1+θ3j−1.The conditional distributions are given by (13) and (14), satisfying the compatibility conditions, and are studied by Arnold et al. [26], which guarantees the existence of the discrete bivariate model (4).The regression functions for these conditional distributions are(16)EJ|K=k=1exp−θ2+θ3k−1,EK|J=j=1exp−θ1+θ3j−1.These regression functions are nonlinear and decreasing (increasing) ifθ3>0 (θ3<0) (see Figure 2).Figure 2
The regression curve ofj on k (k on j) of BPEC distribution for θ3=0.5>0 (θ3=−0.5<0).The first moment of the pair (J, K) is obtained by direct calculations using (4), and we find that(17)EJK=∑j=0∞Nθ1,θ2,θ3−1je−θ1+θ2+θ3je−θ1+eθ3j2.Special Classes. Class (4) can be classified by suitable selections for the parameters θ1,θ2, and θ3 into the following two subclasses.(a)
Subclass I (subclass with two parameters):(18)PJ,Kj,k=Nθ1,θ2,θ3−1expθ1k−θ3jk+θ2j,j,k=0,1,…,θ1,θ2<0,θ3∈ℛ.(1)
θ3=0,J,K are independent, and (4) reduces to(19)PJ,Kj,k=Nθ1,θ2,0−1expθ2jexpθ1k;j,k=0,1,…,θ1<0,θ2∈ℛ,whereNθ1,θ2,0−1=1/eθ1−1eθ2−1.It is clear from (19) that the two random variables (RVs) J and K are independent, with the following marginal densities:(20)PJ|Kj|k=exp−θ2−1exp−θ2j,j,k=0,1,…,(21)PK|Jk|j=exp−θ1−1exp−θ1k,j,k=0,1,…,i.e.,(22)J|K=k∼PEexp−θ2−1,K|J=j∼PEexp−θ1−1.(2)
θ1=θ2, and (4) reduces to(23)PJ,Kj,k=Nθ2,θ2,θ3−1expj+kθ2−θ3jk,j,k=0,1,…,θ2<0,θ3∈ℛ,where(24)Nθ2,θ2,θ3−1=∑x=0∞eθ2+θ3j−eθ2+eθ3j,PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(25)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexp−θ2+θ3j−1.(3)
θ2=0, and (4) becomes(26)PJ,Kj,k=Nθ1,0,θ3−1expθ1k−θ3jk,j,k=0,1,…,θ1<0,θ3∈ℛ,where(27)Nθ1,0,θ3−1=∑x=0∞eθ3j−eθ1+eθ3j,PJ|Kj|k=expθ3k−1expθ3kj,j,k=0,1,…,PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(28)J|K=k∼PEexpθ3k−1,K|J=j∼PEexp−θ1+θ3j−1.(4)
θ1=0, and (4) reduces to(29)PJ,Kj,k=N0,θ2,θ3−1exp−θ3jk+θ2j,j,k=0,1,…,θ2<0,θ3∈ℛ,where(30)N0,θ2,θ3−1=∑k=0∞eθ3k−eθ2+eθ3k,PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,PK|Jk|j=expθ3j−1expθ3jk,j,k=0,1,…,i.e.,(31)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexpθ3j−1.(b)
Subclass II (subclass with one parameter):(1)
θ1=θ2=θ3,θ3>0, and (4) reduces to(32)PJ,Kj,k=Nθ3−1expj−jk+kθ3,j,k=0,1,…,θ3<0,where(33)Nθ3−1=∑y=0∞e2θ3k−eθ3+eθ3k,PJ|Kj|k=expk−1θ3−1expk−1θ3j,j,k=0,1,…,PK|Jk|j=expj−1θ3−1expj−1θ3k,j,k=0,1,…,i.e.,(34)J|K=k∼PEexpk−1θ3−1,K|J=j∼PEexpj−1θ3−1.


## 3.1. Conditional PMF and Moments

The particular manifestations of the conditional distributions to the new model would provide(13)PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,(14)PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(15)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexp−θ1+θ3j−1.The conditional distributions are given by (13) and (14), satisfying the compatibility conditions, and are studied by Arnold et al. [26], which guarantees the existence of the discrete bivariate model (4).The regression functions for these conditional distributions are(16)EJ|K=k=1exp−θ2+θ3k−1,EK|J=j=1exp−θ1+θ3j−1.These regression functions are nonlinear and decreasing (increasing) ifθ3>0 (θ3<0) (see Figure 2).Figure 2
The regression curve ofj on k (k on j) of BPEC distribution for θ3=0.5>0 (θ3=−0.5<0).The first moment of the pair (J, K) is obtained by direct calculations using (4), and we find that(17)EJK=∑j=0∞Nθ1,θ2,θ3−1je−θ1+θ2+θ3je−θ1+eθ3j2.Special Classes. Class (4) can be classified by suitable selections for the parameters θ1,θ2, and θ3 into the following two subclasses.(a)
Subclass I (subclass with two parameters):(18)PJ,Kj,k=Nθ1,θ2,θ3−1expθ1k−θ3jk+θ2j,j,k=0,1,…,θ1,θ2<0,θ3∈ℛ.(1)
θ3=0,J,K are independent, and (4) reduces to(19)PJ,Kj,k=Nθ1,θ2,0−1expθ2jexpθ1k;j,k=0,1,…,θ1<0,θ2∈ℛ,whereNθ1,θ2,0−1=1/eθ1−1eθ2−1.It is clear from (19) that the two random variables (RVs) J and K are independent, with the following marginal densities:(20)PJ|Kj|k=exp−θ2−1exp−θ2j,j,k=0,1,…,(21)PK|Jk|j=exp−θ1−1exp−θ1k,j,k=0,1,…,i.e.,(22)J|K=k∼PEexp−θ2−1,K|J=j∼PEexp−θ1−1.(2)
θ1=θ2, and (4) reduces to(23)PJ,Kj,k=Nθ2,θ2,θ3−1expj+kθ2−θ3jk,j,k=0,1,…,θ2<0,θ3∈ℛ,where(24)Nθ2,θ2,θ3−1=∑x=0∞eθ2+θ3j−eθ2+eθ3j,PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(25)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexp−θ2+θ3j−1.(3)
θ2=0, and (4) becomes(26)PJ,Kj,k=Nθ1,0,θ3−1expθ1k−θ3jk,j,k=0,1,…,θ1<0,θ3∈ℛ,where(27)Nθ1,0,θ3−1=∑x=0∞eθ3j−eθ1+eθ3j,PJ|Kj|k=expθ3k−1expθ3kj,j,k=0,1,…,PK|Jk|j=exp−θ1+θ3j−1exp−θ1+θ3jk,j,k=0,1,…,i.e.,(28)J|K=k∼PEexpθ3k−1,K|J=j∼PEexp−θ1+θ3j−1.(4)
θ1=0, and (4) reduces to(29)PJ,Kj,k=N0,θ2,θ3−1exp−θ3jk+θ2j,j,k=0,1,…,θ2<0,θ3∈ℛ,where(30)N0,θ2,θ3−1=∑k=0∞eθ3k−eθ2+eθ3k,PJ|Kj|k=exp−θ2+θ3k−1exp−θ2+θ3kj,j,k=0,1,…,PK|Jk|j=expθ3j−1expθ3jk,j,k=0,1,…,i.e.,(31)J|K=k∼PEexp−θ2+θ3k−1,K|J=j∼PEexpθ3j−1.(b)
Subclass II (subclass with one parameter):(1)
θ1=θ2=θ3,θ3>0, and (4) reduces to(32)PJ,Kj,k=Nθ3−1expj−jk+kθ3,j,k=0,1,…,θ3<0,where(33)Nθ3−1=∑y=0∞e2θ3k−eθ3+eθ3k,PJ|Kj|k=expk−1θ3−1expk−1θ3j,j,k=0,1,…,PK|Jk|j=expj−1θ3−1expj−1θ3k,j,k=0,1,…,i.e.,(34)J|K=k∼PEexpk−1θ3−1,K|J=j∼PEexpj−1θ3−1.


## 4. Estimation of the Parameters of BPLC

Suppose thatj1,k1,j2,k2,…,jn,kn are random samples from BPECθ1,θ2,θ3 class with density function given in (4).


### 4.1. Maximum Likelihood Estimation (MLE) for the Parameters

The log-likelihood functionlθ of BPECθ1,θ2,θ3 is given by(35)lθ=−nlogNθ1,θ2,θ3+θ1∑i=1nki−θ3∑i=1njiki+θ2∑i=1nji.The maximum likelihood estimates ofθ1,θ2, and θ3 can be obtained by solving(36)∂Nθ/∂θ1Nθ=1n∑i=1nji,(37)∂Nθ/∂θ2Nθ=1n∑i=1nji,(38)∂Nθ/∂θ3Nθ=−1n∑i=1njiki,where Nθ is given by (12).The implicit nature of systems (36)–(38) suggests the numerical derivation of the MLE of parameters θ1,θ2, and θ3.


### 4.2. Pseudolikelihood Estimation for the Parameters

The pseudolikelihood method is an alternative estimation technique that does not include the normalizing constant (see Besag [34, 35] and Arnold and Strauss [36, 37]). The pseudolikelihood function can be written as(39)PLθ=∏i=1nPj|kji|kiPk|jki|ji,PLθ=∏i=1nexp−θ2+θ3k−1exp−θ2+θ3kjexp−θ1+θ3j−1exp−θ1+θ3jk.Therefore, we have the following logarithmic form of the pseudolikelihood function:(40)logPLθ=∑i=1nlogexp−θ2+θ3ki−1−∑i=1nji−θ2+θ3ki+∑i=1nlogexp−θ1+θ3ji−1−∑i=1nki−θ1+θ3ji.The maximum pseudolikelihood estimates ofθ1,θ2, and θ3 can be obtained by solving the following:(41)∂logPLθ1,θ2,θ3∂θ1=−∑i=1nexp−θ1+θ3jiexp−θ1+θ3ji−1+∑i=1nki,∂logPLθ1,θ2,θ3∂θ2=−∑i=1nexp−θ2+θ3kiexp−θ2+θ3ki−1+∑i=1nxi,∂logPLθ1,θ2,θ3∂θ3=∑i=1njiexp−θ1+θ3jiexp−θ1+θ3ji−1+∑i=1nkiexp−θ2+θ3kiexp−θ2+θ3ki−1+2∑i=1njiki.


## 4.1. Maximum Likelihood Estimation (MLE) for the Parameters

The log-likelihood functionlθ of BPECθ1,θ2,θ3 is given by(35)lθ=−nlogNθ1,θ2,θ3+θ1∑i=1nki−θ3∑i=1njiki+θ2∑i=1nji.The maximum likelihood estimates ofθ1,θ2, and θ3 can be obtained by solving(36)∂Nθ/∂θ1Nθ=1n∑i=1nji,(37)∂Nθ/∂θ2Nθ=1n∑i=1nji,(38)∂Nθ/∂θ3Nθ=−1n∑i=1njiki,where Nθ is given by (12).The implicit nature of systems (36)–(38) suggests the numerical derivation of the MLE of parameters θ1,θ2, and θ3.


## 4.2. Pseudolikelihood Estimation for the Parameters

The pseudolikelihood method is an alternative estimation technique that does not include the normalizing constant (see Besag [34, 35] and Arnold and Strauss [36, 37]). The pseudolikelihood function can be written as(39)PLθ=∏i=1nPj|kji|kiPk|jki|ji,PLθ=∏i=1nexp−θ2+θ3k−1exp−θ2+θ3kjexp−θ1+θ3j−1exp−θ1+θ3jk.Therefore, we have the following logarithmic form of the pseudolikelihood function:(40)logPLθ=∑i=1nlogexp−θ2+θ3ki−1−∑i=1nji−θ2+θ3ki+∑i=1nlogexp−θ1+θ3ji−1−∑i=1nki−θ1+θ3ji.The maximum pseudolikelihood estimates ofθ1,θ2, and θ3 can be obtained by solving the following:(41)∂logPLθ1,θ2,θ3∂θ1=−∑i=1nexp−θ1+θ3jiexp−θ1+θ3ji−1+∑i=1nki,∂logPLθ1,θ2,θ3∂θ2=−∑i=1nexp−θ2+θ3kiexp−θ2+θ3ki−1+∑i=1nxi,∂logPLθ1,θ2,θ3∂θ3=∑i=1njiexp−θ1+θ3jiexp−θ1+θ3ji−1+∑i=1nkiexp−θ2+θ3kiexp−θ2+θ3ki−1+2∑i=1njiki.


## 5. Application

We consider a dataset in this paper which was obtained from Mitchell and Paulson [38] and is presented in Table 1. Utilizing these data, we should gauge and estimate the parameters θ1,θ2, and θ3 of class (4). The information includes flight aborts count data from 109 aircrafts, and the variables J and K represent the flight aborts in the first and second sequential six months of a one-year period.Table 1
The flight abort check information starting with 109 aircraft.
jK01234∑034204646811770002426410011304000440000005200002∑5935564109The frequencies of the observed data provide several (j, 0) and (0, k) data, indicating a negative correlation between j and k. Therefore, we fit BPC, BP, BPL, and BNB distributions to the data since these distributions can be fitted to bivariate data with positive, zero, or negative correlation.The statistic measures for the given data arej¯=0.62, s12=1.03; k¯=0.72, s22=1.07, Covj,k=−0.169, and ρ=−0.16. Table 2 presents the estimated parameters of the BPEC model and its mean square error (MSE).The joint PMF of bivariate Poisson conditional distribution can be defined asBPCλ1,λ2,λ3 and defined as follows (Arnold and Strauss [36]):(42)PJ=j,K=k=Cλ1,λ2,λ3λ1jλ2kλ3jkj!k!,j,k=0,1,2,…,λ1,λ2>0,0<λ3≤1,where Cλ1,λ2,λ3 is constant. The conditionals K|J and J|K are λ2λ3j and λ1λ3k, respectively.Table 2
Estimation of parameters of the BPEC.
ParametersMLEMSEMPLEMSEθ1−0.67120.271675−0.9324980.000506142θ2−0.76640.0709704−0.8243660.000593723θ3−1.64370.02065830.570960.00167806The joint PMF of BPλ1,λ2,α distribution is (Lakshminarayana et al. [39])(43)PJ=j,K=k=e−λ1−λ2λ1jλ2kjk1+αe−j−e−cλ1e−k−e−cλ2,j,k=0,1,2,…,λ1,λ2>0,where c=1−e−1, EJ=VarJ=λ1,EK=VarK=λ2,CovJ,K=αλ1λ2c2e−cλ1+λ2, and α can be chosen such that PY1,Y2 will be the PMF.The joint PMF ofBPLθ1,θ2,m1,m2,γ distribution is (Zamani et al. [40])(44)PJ=j,K=k=θ12j+θ1+2θ12j+θ1+2θ1+1j+3θ2+1k+31+αe−j−c1e−k−c2,j,k=0,1,2,…,θ1,θ2>0,where(45)c1=Ee−J=θ121+θ1θ1+2−e−1θ1−e−1+12,c2=Ee−K=θ221+θ2θ2+2−e−1θ2−e−1+12.The joint PMF ofBNBθ1,θ2,m1,m2,γ distribution is (Famoye [41])(46)PJ=j,K=k=m1−1+j−1kθ1j1−θ1m1−1m2−1+j−1kθ2k1−θ2m2−11+γe−j−c1e−k−c2,j,k=0,1,2,…,θ1,θ2>0,0<m1,m2<1,where(47)c1=Ee−J=1−θ11−θ1e−1m1−1,c2=Ee−K=1−θ21−θ2e−1m2−1,and the mean, variance, and covariance are(48)EJ=m1−1θ11−θ1,EK=m2−1θ21−θ2,VarJ=m1−1θ11−θ12,VarK=m2−1θ21−θ22,CovJ,K=γc1c2∏imαi−1θie−11−θie−1−mi−1θi1−θi.We used the Mathematica package to estimate the parameters of BPEC distribution.The new distribution BPEC is more appropriate as we can see in Table3 as compared to the BPC, BP, BPL, and BNB distributions, where the BPEC distribution gives the largest value for the AIC and BIC statistics compared to other models.Table 3
Parameter estimators and AIC and BIC of BPEC, BPC, BP, BNB, and BPL distributions.
ModelParameterMLELog-likelihoodAICBICBPECθ1−0.6712−114.821−120.821−121.858θ2−0.7664θ3−1.6437BPCλ10.14388−252.078−258.078−259.115λ20.208112λ30.426338BPλ10.6129−254.99−260.99−262.027λ20.7131α−0.9290BPLθ12.1165−244.62−250.62−251.657θ21.8607α−1.0363BNBθ10.4045−244.27−254.27−255.998θ20.3138m11.0977m20.6305γ−1.1109


## 6. Conclusion

In this work, a BPEC model is presented by determining conditional discrete Poisson exponential distributions. Therefore, we obtained the statistical properties and special classes for BPEC distribution. The estimation of BPEC parameters through the techniques for MLE and MPLE is presented. In view of the findings presented in Table1, the MPLE is better than MLE because the MPLE technique uses conditional distributions which in our case do not suffer from the problem caused by the normalizing constant. Moreover, the AIC and BIC depict that BPEC distribution adequately fits the considered dataset compared to the BPC, BP, BPL, and BNB distributions.



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