A New Model of Discrete-Continuous Bivariate Distribution with Applications to Medical Data

The bivariate Poisson exponential-exponential distribution is an important lifetime distribution in medical data analysis. In this article, the conditionals, probability mass function (pmf), Poisson exponential and probability density function (pdf), and exponential distribution are used for creating bivariate distribution which is called bivariate Poisson exponential-exponential conditional (BPEEC) distribution. Some properties of the BPEEC model are obtained such as the normalized constant, conditional densities, regression functions, and product moment. Moreover, the maximum likelihood and pseudolikelihood methods are used to estimate the BPEEC parameters based on complete data. Finally, two data sets of real bivariate data are analyzed to compare the methods of estimation. In addition, a comparison between the BPEEC model with the bivariate exponential conditionals (BEC) and bivariate Poisson exponential conditionals (BPEC) is considered.


Introduction
The construction of bivariate distributions based on specifying the marginals or the conditionals received great attention from researchers since the beginning of the nineties [1][2][3][4]. Some conferences held about this technique, as the one was held in New York (Fisher and Sen [5]) under the title "The collected works of Wassily Hoeffding," and in Prague (Benes and Stepan [6]) under the title "Distributions with given marginals and moment problems." It is often easier to visualize conditional pdf (pmf) or properties of f X|Y ðx | yÞ andf Y|X ðy | xÞ then f x ðxÞandf y ðyÞ or f X,Y ðx, yÞ [7] (p.1, [8]) rather than the joint distribution. In this sense, studying the bivariate distributions when both conditionals belong to discrete or continuous distributions has received special attention, as Arnold et al. [7,9], Johnson et al. [10], and Castillo and Galambos [11][12][13] see also Arnold [14], Arnold et al. [15] Kottas et al. [16], Gharib et al. [17], and Mohammed et al. [4,18]. But what is new in this paper is that one distribution is discrete and the other is continuous in the considered bivariate distributions. A similar type of class was derived by Sarabia et al. [19] where the conditional distributions of those bivariate distributions were Poisson (discrete) and gamma (continuous). The utilization of this type of class in bonusmalus systems (Sarabia et al. [19]), medical applications as analysis of HIV infection (Nazife Sultanoglu et al. [20]), and fractional modeling for improving the scholastic performance of students with optimal control Abdullahi (Yusuf et al. [21]).
In this paper, we will be interested in studying an interesting trend in constructing a set of bivariate distributions with discrete and continuous conditional distributions as exponential and exponential Poisson distributions, respectively.

Bivariate Poisson Exponential-Exponential Conditionals (BPEEC) Class
Suppose that the conditional distributions X | Y and Y | X are, respectively, where λðyÞ and βðxÞ are some positive functions. According to these conditional distributions, the f X,Y ðx, yÞ is where f X ðxÞ and f Y ðyÞ are, respectively, the marginal distributions of X and Y. Then, Considering By substituting equations (6) and (7) into equation (5) we obtain equation Equation (8) is functional equation, which is a special of ∑ n k=1 f k ðxÞg k ðyÞ = 0, whose general solution is given by Aczel [24], (p. 161), as substituting (9) into (5) results in where The joint distribution f X,Y ðx, yÞ in equation (10) describes the new model of BPEEC distribution that has α 1 ðα 2 Þ intensity parameters for X (Y) and α 3 dependence parameter, where α 3 = 0 coincide with independence between X and Y.

Properties of BPEEC Class
The general properties of BPEEC class are studied in this part.
3.1. Normalizing Constant. The normalizing constant ½NðΑÞ −1 of the discrete-continuous BPEEC class given in equation (10) is The previous expression could be written in a new form: Therefore, given that, Both f X|Y ðx | yÞ and f Y|X ðy | xÞ given in equations (14) and (15) are satisfying the compatibility conditions stated by Arnold et al. [7], for the existing BPEEC class in equation (10).
3 Computational and Mathematical Methods in Medicine where HypergeometricPFQ ½fa 1 , ⋯, a p g, fb 1 , ⋯, b q g, z is the generalized hypergeometric function p F q ða ; b ; zÞ.

Parameter Estimation
In the section, the maximum likelihood estimation (MLE) and maximum pseudolikelihood estimator (MPLE) are used to estimate α 1 , α 2 and α 3 of BPEEC class.
4.1. The Maximum Likelihood Estimation. Suppose that ðx i , y i Þ, ði = 1, 2, ⋯, nÞ are observed values from the BPEEC distribution with f X,Y ðx, yÞ given in equation (10), then the logarithm of the likelihood function is The estimates of α 1 , α 2 and α 3 are obtained by differentiating lðΑÞ with respect to each parameter. This results in the following likelihood equations: Solving the previous system of nonlinear equations ∂lðΑÞ/∂α i j α i = b α i = 0, i = 1, 2, 3, means identifying estimated values for b α 1 , b α 2 and b α 3 . This can be done numerically either using finite difference methods (Abu Arqub and Abo-Hammour [25]) or using optimization techniques and calculating the minimum residual error (Abo-Hammour et al. [26,27]).

Kidney Infection Data.
The bivariate data set in Table 2 represents the infection for kidney patients and has been obtained from Gilchrist and Aisbett [30]. Let X and Y be the first and second recurrence times, respectively. From the obtained results of previous cases, the AIC and BIC of the BPEEC model are more than the corresponding of the BPEC and BEC models which means that the BPEEC model is a better fit for the given data. The approximated 95% two-sided CI of the parameters α 1 , α 2 and α 3 are given, respectively, as  Tables 5 and 6 as compared to the other models.

Conclusion
In this article, a bivariate Poisson exponential-exponential distribution (BPEEC) is introduced by specified conditional pmf and pdf distributions as Poisson exponential and exponential distributions, respectively. In addition, we obtained some properties such as conditional, marginal distributions, and moments. The MLE and MPLE of α 1 , α 2 , and α 3 for       Tables 5 and 6, the model selection AIC and BIC of discrete-continuous BPEEC distribution are better than the discrete BPEC and continuous BEC distributions We will apply and investigate the effectiveness of the proposed BPEEC model in censored experiments either on simulation studies or in different real-world scenarios.

Data Availability
All data are available in the paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest.