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

Authors: B. I. Mohammed; Nicholas Makumi; Ramy Aldallal; Taysir E. Dyhoum; Hassan M. Aljohani
Journal: Computational and Mathematical Methods in Medicine
 (2022)

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## Abstract

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.


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## Body

## 1. 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 [1–4]. 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 fX∣Yx∣yandfY∣Xy∣x then fxxandfyy or fX,Yx,y [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 [11–13] 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 bonus-malus 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.The univariate discrete Poisson exponential distribution has the following pmf (Fazal and Bashir [22]):
(1)PX=x=λ1+λx,x=1,2,⋯,λ>0.i.e.,X~PEλ. Moreover, the univariate continuous exponential distribution has the following pdf (Fazal and Bashir [23]):
(2)fyy=βy−β,y>0,β>0.i.e.,Y~Eβ.


## 2. Bivariate Poisson Exponential-Exponential Conditionals (BPEEC) Class

Suppose that the conditional distributionsX∣Y and Y∣X are, respectively,
(3)fXYx∣y=λy1+λyx+1,i.e.XY=y~Poissonexponentialλy.fYXy∣x=βxExp−βxy,i.e.YX=x~Exponentialβx,whereλy and βx are some positive functions.According to these conditional distributions, thefX,Yx,y is
(4)λy1+λyx+1fYy=βxExp−βxyfXx,where fXx and fYy are, respectively, the marginal distributions of X and Y. Then,
(5)logλyfYy−x+1log1+λy=logβxfXx−βxy.Considering(6)hy=logλyfYy,(7)gx=logβxfXx.By substituting equations (6) and (7) into equation (5) we obtain equation
(8)gx−βxy−hy+x+1log1+λy=0.Equation (8) is functional equation, which is a special of ∑k=1nfkxgky=0, whose general solution is given by Aczel [24], (p. 161), as
(9)βx=α2−α3x+1,Log1+λy=−α1−α3ygx=−α0+α1x+1,hy=−α0−α2y,substituting (9) into (5) results in
(10)fX,Yx,y=NΑ−1Expα1x+1−α2−α3x+1y,α1>0,α2,α3∈ℝ,x=0,1,2,⋯,y>0,where NΑ−1=Exp−α0,Α=α1,α2,α3, is the normalizing constant.The joint distributionfX,Yx,y in equation (10) describes the new model of BPEEC distribution that has α1α2 intensity parameters for X (Y) and α3 dependence parameter, where α3=0 coincide with independence between X and Y.


## 3. Properties of BPEEC Class

The general properties of BPEEC class are studied in this part.


### 3.1. Normalizing Constant

The normalizing constantNΑ−1 of the discrete-continuous BPEEC class given in equation (10) is
(11)NΑ=∑x=0∞∫0∞Expα1x+1−α2y+α3x+1ydy=∑x=0∞eα1x+1α2−α3x+1.The previous expression could be written in a new form:(12)1α2−α3x+1=1α21−α3α2x+1−1=1α2∑k=0∞α3α2x+1k.Therefore,(13)NΑ=1α2∑k=0∞α3α2k∑x=0∞x+1kExpα1x+1=1α2∑k=0∞α3α2kdkdα1kExpα11−Expα1.


### 3.2. Conditional Distributions and Regression Functions

A specific form of thefX∣Yx∣y and fY∣Xy∣x for the new model can be define as
(14)fX∣Yx∣y=Exp−α1−α3y−1Exp−α1−α3yx+1,x=0,1,2,⋯,y>0,(15)fY∣Xy∣x=α2−α3x+1Expα2−α3x+1y,x=0,1,2,⋯,y>0,given that,(16)X∣Y=y~PoissonExponentialExp−α1−α3y−1,Y∣X=x~Expα2−α3x+1.BothfX∣Yx∣y and fY∣Xy∣x 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).The regression functions offXYx∣y and fYXy∣x, are, respectively, defined as
(17)EXY=y=1Exp−α1−α3y−1,EYX=x=1α2−α3x+1.These functions are nonlinear, and we noticed thatEXY=y and EYX=x are decreasing if θ3<0 and increasing if θ3>0. Figures 1 and 2 demonstrate that.Figure 1
TheEXY=y curve of the BPEEC distribution given by equation (10).Figure 2
TheEYX=x curve of the BPEEC distribution given by equation (10).


### 3.3. Marginals and Moments

We substitute (9) into (6) and (7) to get the following marginal functions
(18)fXx=NΑ−1α2−α3x+1Expα1x+1,fYy=NΑ−1Expα1−α3y−1Exp−α2y.For1+x<θ2/θ3, the product-moment of BPEEC distribution is
(19)EXY=NΑ−1ⅇ2α1HypergeometricPFQ2,2−α2/α3,2−α2/α3,3−α2/α3,3−α2/α3,ⅇα1α2−2α32,where HypergeometricPFQa1,⋯,ap,b1,⋯,bq,z is the generalized hypergeometric function Fpqa;b;z.


## 3.1. Normalizing Constant

The normalizing constantNΑ−1 of the discrete-continuous BPEEC class given in equation (10) is
(11)NΑ=∑x=0∞∫0∞Expα1x+1−α2y+α3x+1ydy=∑x=0∞eα1x+1α2−α3x+1.The previous expression could be written in a new form:(12)1α2−α3x+1=1α21−α3α2x+1−1=1α2∑k=0∞α3α2x+1k.Therefore,(13)NΑ=1α2∑k=0∞α3α2k∑x=0∞x+1kExpα1x+1=1α2∑k=0∞α3α2kdkdα1kExpα11−Expα1.


## 3.2. Conditional Distributions and Regression Functions

A specific form of thefX∣Yx∣y and fY∣Xy∣x for the new model can be define as
(14)fX∣Yx∣y=Exp−α1−α3y−1Exp−α1−α3yx+1,x=0,1,2,⋯,y>0,(15)fY∣Xy∣x=α2−α3x+1Expα2−α3x+1y,x=0,1,2,⋯,y>0,given that,(16)X∣Y=y~PoissonExponentialExp−α1−α3y−1,Y∣X=x~Expα2−α3x+1.BothfX∣Yx∣y and fY∣Xy∣x 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).The regression functions offXYx∣y and fYXy∣x, are, respectively, defined as
(17)EXY=y=1Exp−α1−α3y−1,EYX=x=1α2−α3x+1.These functions are nonlinear, and we noticed thatEXY=y and EYX=x are decreasing if θ3<0 and increasing if θ3>0. Figures 1 and 2 demonstrate that.Figure 1
TheEXY=y curve of the BPEEC distribution given by equation (10).Figure 2
TheEYX=x curve of the BPEEC distribution given by equation (10).


## 3.3. Marginals and Moments

We substitute (9) into (6) and (7) to get the following marginal functions
(18)fXx=NΑ−1α2−α3x+1Expα1x+1,fYy=NΑ−1Expα1−α3y−1Exp−α2y.For1+x<θ2/θ3, the product-moment of BPEEC distribution is
(19)EXY=NΑ−1ⅇ2α1HypergeometricPFQ2,2−α2/α3,2−α2/α3,3−α2/α3,3−α2/α3,ⅇα1α2−2α32,where HypergeometricPFQa1,⋯,ap,b1,⋯,bq,z is the generalized hypergeometric function Fpqa;b;z.


## 4. Parameter Estimation

In the section, the maximum likelihood estimation (MLE) and maximum pseudolikelihood estimator (MPLE) are used to estimateα1,α2 and α3 of BPEEC class.


### 4.1. The Maximum Likelihood Estimation

Suppose thatxi,yi,i=1,2,⋯,n are observed values from the BPEEC distribution with fX,Yx,y given in equation (10), then the logarithm of the likelihood function is
(20)lΑ=−nlogNΑ+∑i=1nα1x+1−α2−α3x+1y.The estimates ofα1,α2 and α3 are obtained by differentiating lΑ with respect to each parameter. This results in the following likelihood equations:
(21)∂lΑ∂α1=−nNΑ∂NΑ∂α1+∑i=1nxi+1,∂lΑ∂α2=−nNΑ∂NΑ∂α2−∑i=1nyi,∂lΑ∂α3=−nNΑ∂NΑ∂α3+∑i=1nxi+1yi.Solving the previous system of nonlinear equations∂lΑ/∂αiαi=αi^=0,i=1,2,3, means identifying estimated values for α^1,α^2 and 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]).The normal approximation of the MLE can be also used to create asymptotic confidence intervals (Cls) forαi,i=1,2,3 when the sample size is large. A two-sided (1-α) 100% of CIs for αi,i=1,2,3 are defined as αi^±Zα/2Varαi^, where Varαi^, i=1,2,3 are the asymptotic variances of αi^.


### 4.2. Maximum Pseudolikelihood Estimator

Benes and Stepan and Besag [6, 28] and Arnold and Strauss [2] discussed the MPLE to estimate all model parameters of bivariate distributions. The pseudolikelihood function depends on fX∣Yx∣y and fY∣Xy∣x and does not contain the normalizing constant. Hence, the pseudolikelihood function is
(22)PLΑ=∏i=1nfxyxi∣yifyxyi∣xi.Wheneverxi,yi,i=1,2,⋯,n are observed from the BPEEC class with fX,Yx,y stated in equation (10), the log pseudolikelihood function will be given by
(23)logPLΑ=∑i=1nlogExp−α1−α3yi−1−∑i=1n−α1−α3yixi+1+∑i=1nlogα2−α3xi+1−∑i=1nα2−α3xi+1yi.The estimates ofα1,α2 and α3 are obtained by differentiating the logPLΑ with respect to α1,α2 and α3, respectively, and then, the pseudolikelihood equations are given by
(24)∂logPLΑ∂α1=−∑i=1nExp−α1−α3yiExp−α1−α3yi−1+∑i=1nxi+1−α1−α3yixi,∂logPLΑ∂α3=−∑i=1nyiExp−α1−α3yiExp−α1−α3yi−1+∑i=1nxi+1−α1−α3yixiyi+∑i=1nxi+1α2−α3xi+1−∑i=1nxi+1yi.Solving nonlinear equations∂logPLΑ/∂αiαi=αi^=0,i=1,2,3, the MPLEs α^1,α^2 and α^3 can be obtained using Mathematica package.


## 4.1. The Maximum Likelihood Estimation

Suppose thatxi,yi,i=1,2,⋯,n are observed values from the BPEEC distribution with fX,Yx,y given in equation (10), then the logarithm of the likelihood function is
(20)lΑ=−nlogNΑ+∑i=1nα1x+1−α2−α3x+1y.The estimates ofα1,α2 and α3 are obtained by differentiating lΑ with respect to each parameter. This results in the following likelihood equations:
(21)∂lΑ∂α1=−nNΑ∂NΑ∂α1+∑i=1nxi+1,∂lΑ∂α2=−nNΑ∂NΑ∂α2−∑i=1nyi,∂lΑ∂α3=−nNΑ∂NΑ∂α3+∑i=1nxi+1yi.Solving the previous system of nonlinear equations∂lΑ/∂αiαi=αi^=0,i=1,2,3, means identifying estimated values for α^1,α^2 and 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]).The normal approximation of the MLE can be also used to create asymptotic confidence intervals (Cls) forαi,i=1,2,3 when the sample size is large. A two-sided (1-α) 100% of CIs for αi,i=1,2,3 are defined as αi^±Zα/2Varαi^, where Varαi^, i=1,2,3 are the asymptotic variances of αi^.


## 4.2. Maximum Pseudolikelihood Estimator

Benes and Stepan and Besag [6, 28] and Arnold and Strauss [2] discussed the MPLE to estimate all model parameters of bivariate distributions. The pseudolikelihood function depends on fX∣Yx∣y and fY∣Xy∣x and does not contain the normalizing constant. Hence, the pseudolikelihood function is
(22)PLΑ=∏i=1nfxyxi∣yifyxyi∣xi.Wheneverxi,yi,i=1,2,⋯,n are observed from the BPEEC class with fX,Yx,y stated in equation (10), the log pseudolikelihood function will be given by
(23)logPLΑ=∑i=1nlogExp−α1−α3yi−1−∑i=1n−α1−α3yixi+1+∑i=1nlogα2−α3xi+1−∑i=1nα2−α3xi+1yi.The estimates ofα1,α2 and α3 are obtained by differentiating the logPLΑ with respect to α1,α2 and α3, respectively, and then, the pseudolikelihood equations are given by
(24)∂logPLΑ∂α1=−∑i=1nExp−α1−α3yiExp−α1−α3yi−1+∑i=1nxi+1−α1−α3yixi,∂logPLΑ∂α3=−∑i=1nyiExp−α1−α3yiExp−α1−α3yi−1+∑i=1nxi+1−α1−α3yixiyi+∑i=1nxi+1α2−α3xi+1−∑i=1nxi+1yi.Solving nonlinear equations∂logPLΑ/∂αiαi=αi^=0,i=1,2,3, the MPLEs α^1,α^2 and α^3 can be obtained using Mathematica package.


## 5. Applications

### 5.1. Seizure Data

The bivariate data set in Table1 has been obtained from Johnson and Davis [29]. This data represents the number of seizures observed in the first week and the second week for 30 patients after admission to the hospital.Table 1
Seizure data for 30 patients.
Patient12345678910Week-1(X)5113301403Week-2 (Y)0242100002Patient11121314151617181920Week-1(X)3313011012Week-2 (Y)0202003211Patient21222324252627282930Week-1(X)0211020163Week-2 (Y)0140020100The joint pmf ofBPECθ1,θ2,θ3 distribution is Mohammed et. al [18])
(25)PX,Kx,y=Nθ1,θ2,θ3−1expθ1y−θ3xy+θ2x,x,y=1,⋯,θ1,θ2<0,θ3∈R,where Nθ1,θ2,θ3−1 is the normalizing constant. The conditionals Y∣X and X∣Y are Poisson exponential distributions.The joint pdf ofBECλ1,λ2,λ3 distribution is Arnold and Strauss [2])
(26)fX,Yx,y=Nλ1,λ2,λ3−1exp−λ1y−λ2x+λ3xy,x>0,y>0,λ1>0,λ2>0,λ3≤0,where Nλ1,λ2,λ3−1 is the normalizing constant. The conditionals YX and XY are exponential distributions.


### 5.2. Kidney Infection Data

The bivariate data set in Table2 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.Table 2
Kidney infection data for 38 patients.
Patient12345678910Xi82322447302475115315Yi16132831812245930196154Patient11121314151617181920Xi714196149536171852922215Yi33383870254117114159108Patient21222324252627282930Xi152402133912113132342130Yi36224664640201156302526Patient3132333435363738Xi27515219011954663Yi5843305816788From 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 parametersα1,α2 and α3 are given, respectively, as [0.0287,0.1555], [−0.0419,1.1271], and [−0.44599, 0.14019] for seizure data, whilst for the Kidney infection data are [0.02087,0.16351], [−0.11518,1.2005], and [−0.48284,0.1769]. Tables 3 and 4 present the estimated parameters of the BPPEC model and its mean square error (MSE). The BPPEC model is more appropriate as we can see in Tables 5 and 6 as compared to the other models.


## 5.1. Seizure Data

The bivariate data set in Table1 has been obtained from Johnson and Davis [29]. This data represents the number of seizures observed in the first week and the second week for 30 patients after admission to the hospital.Table 1
Seizure data for 30 patients.
Patient12345678910Week-1(X)5113301403Week-2 (Y)0242100002Patient11121314151617181920Week-1(X)3313011012Week-2 (Y)0202003211Patient21222324252627282930Week-1(X)0211020163Week-2 (Y)0140020100The joint pmf ofBPECθ1,θ2,θ3 distribution is Mohammed et. al [18])
(25)PX,Kx,y=Nθ1,θ2,θ3−1expθ1y−θ3xy+θ2x,x,y=1,⋯,θ1,θ2<0,θ3∈R,where Nθ1,θ2,θ3−1 is the normalizing constant. The conditionals Y∣X and X∣Y are Poisson exponential distributions.The joint pdf ofBECλ1,λ2,λ3 distribution is Arnold and Strauss [2])
(26)fX,Yx,y=Nλ1,λ2,λ3−1exp−λ1y−λ2x+λ3xy,x>0,y>0,λ1>0,λ2>0,λ3≤0,where Nλ1,λ2,λ3−1 is the normalizing constant. The conditionals YX and XY are exponential distributions.


## 5.2. Kidney Infection Data

The bivariate data set in Table2 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.Table 2
Kidney infection data for 38 patients.
Patient12345678910Xi82322447302475115315Yi16132831812245930196154Patient11121314151617181920Xi714196149536171852922215Yi33383870254117114159108Patient21222324252627282930Xi152402133912113132342130Yi36224664640201156302526Patient3132333435363738Xi27515219011954663Yi5843305816788From 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 parametersα1,α2 and α3 are given, respectively, as [0.0287,0.1555], [−0.0419,1.1271], and [−0.44599, 0.14019] for seizure data, whilst for the Kidney infection data are [0.02087,0.16351], [−0.11518,1.2005], and [−0.48284,0.1769]. Tables 3 and 4 present the estimated parameters of the BPPEC model and its mean square error (MSE). The BPPEC model is more appropriate as we can see in Tables 5 and 6 as compared to the other models.


## 6. 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 ofα1,α2, and α3 for BPEEC distribution are present. By analyzing the results obtained, we get the following:
(i)
From Tables3 and 4, the results of MPLE are better than MLE since the MPLE method does not contain a normalizing constant(ii)
From Tables5 and 6, the model selection AIC and BIC of discrete-continuous BPEEC distribution are better than the discrete BPEC and continuous BEC distributionsTable 3
Parameter’s estimation of the BPEEC.
True valuesMLEMSEMPLEMSEα1=0.50.09210.35070.32930.0291α2=0.60.54260.00330.60740.0001α3=0.5−0.15290.42640.57580.0057Table 4
Parameters’ estimation of the BPEEC.
True valuesMLEMSEMPLEMSEθ1=0.80.62550.03040.56510.0042θ2=0.30.01150.0832−0.34970.4221θ3=0.50.29990.04000.39970.0101Table 5
Log-likelihood, AIC, and BIC of BPEEC, BPEC, and BEC distributions.
ModelParameterMLELog-likelihoodAICBICBPEECα10.09219−44.647−50.647−49.749α20.54266α3−0.15297BPECθ1−0.53262−95.895−101.895−100.997θ2−0.40667θ30.04571BECλ1−0.98912−76.485−82.485−81.5876λ20.56715λ30.01339Table 6
Log-likelihood, AIC, and BIC of BPEEC, BPEC, and BEC distributions.
ModelParameterMLELog-likelihoodAICBICBPEECα10.62556122711122704.8122705.3α2−0.30913α30.29990BPECθ1−0.011444065.14059.14059.6θ2−0.00510θ3−0.01135BECλ1−0.00357−444.99−450.99−450.45λ20.00078λ30.00007We will apply and investigate the effectiveness of the proposed BPEEC model in censored experiments either on simulation studies or in different real-world scenarios.



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