The Bivariate Poisson–X–Exponential Distribution: Theory, Inference, and Multidomain Applications

Abstract

We propose the Bivariate Poisson–X–Exponential Distribution (BPXED), a flexible bivariate count model obtained by compounding Poisson variables with a shared X–Exponential latent mixing distribution. The model extends the Poisson–X–Exponential (PXED) distribution and includes several bivariate Poisson-type models as special or limiting cases. Closed-form expressions are derived for the joint probability mass function, probability generating function, moments, and covariance structure, showing that dependence arises from shared latent heterogeneity and is restricted to positive correlation. Parameter estimation is developed using maximum likelihood, regression-based, and Bayesian approaches, and a Monte Carlo simulation study demonstrates a good finite-sample performance. Applications to soccer scores, reliability failures, and correlated photon counts illustrate improved goodness-of-fit over classical and recent competing models. Overall, BPXED provides an analytically tractable and interpretable framework for modeling positively dependent and overdispersed bivariate count data.

1. Introduction

Count data modeling plays a central role in applied statistics and arises in diverse fields such as reliability engineering, epidemiology, environmental science, and sports analytics. The Poisson distribution remains the most classical and parsimonious model for count data; however, its fundamental assumption of equidispersion—equality of the mean and variance—is often violated in practice. Empirical datasets frequently exhibit overdispersion and latent heterogeneity, which motivates the development of more flexible count models.

One of the most successful strategies for accommodating overdispersion is the Poisson mixture (or compounding) framework. This approach has generated several influential distributional families, including the Poisson–Gamma (negative binomial), Poisson–Lindley [1], and Poisson–Lindley–Quasi XGamma [2] distributions. Further extensions, such as the new three-parameter Poisson–Lindley model [3], have expanded this framework by allowing finer control over dispersion and capturing more intricate variability patterns encountered in reliability and risk analysis.

Building on this line of research, Yousfi and Zeghdoudi [4] propose the Poisson–X–Exponential (PXED) distribution by incorporating the X–Exponential kernel into the Poisson mixing structure, thereby addressing overdispersion while preserving analytical tractability. In the multivariate context, Arrar et al. [5] propose the Bivariate Poisson–XLindley (BPXL) distribution with applications to soccer match outcomes, whereas Haddari et al. [6] develop the Modified Bivariate Poisson–Lindley model (BPNXLD), offering enhanced flexibility in modeling dependence structures. These developments are closely related to reliability modeling under exponential-type distributions, where Bayesian estimation techniques have been explored under various loss functions [7]. Moreover, the application of bivariate Poisson models to sports data analysis has been well established in the literature, particularly in modeling match scores and dependence between competing teams [8].

Beyond mixture-based constructions, alternative mechanisms have been proposed to introduce dependence in bivariate count models. For example, Genest et al. [9] developed a bivariate Poisson common-shock model capable of generating a wide range of positive dependence through a shared latent shock component. Using a conditional specification approach, Ghosh et al. [10] introduced a tractable bivariate Poisson model with explicit joint and conditional structures. From a compound distribution perspective, Abdelghani et al. [11] proposed a bivariate model based on Poisson maxima of Gamma variates, providing additional flexibility in tail behavior. More recently, Maya et al. [12] developed bivariate Poisson extended exponential distributions together with associated BINAR(1) processes, enabling joint modeling of contemporaneous and temporal dependence in multivariate count time series. Collectively, these contributions highlight the diversity of strategies used to relax the restrictive assumptions of the classical bivariate Poisson model while maintaining mathematical and computational tractability.

Despite these advances, many existing models are motivated primarily by specific application domains or rely predominantly on Lindley-type mixing structures. Moreover, comprehensive evaluations of a single bivariate count model across substantially different scientific fields remain scarce.

In this context, the present work introduces a unified bivariate Poisson-type model designed for broad applicability across multiple domains. To the best of our knowledge, few studies have systematically assessed the same bivariate count model in diverse applied settings such as:

• Sports analytics (e.g., soccer goal modeling and outcome prediction);
• Reliability engineering (e.g., dependent system failures); and
• Astronomy and physics (e.g., correlated photon-count data).

We propose a new bivariate discrete distribution—the Bivariate Poisson–X–Exponential Distribution (BPXED)—constructed by compounding two Poisson variables with a shared latent X–Exponential mixing variable. This formulation induces positive dependence through a common latent factor while permitting flexible marginal dispersion. The BPXED model encompasses the classical Bivariate Poisson (BP) and the Bivariate Poisson–Lindley (BPLD) as special or limiting cases, thereby offering enhanced interpretability and dispersion control within a coherent and analytically tractable framework.

The remainder of the paper is organized as follows. Section 2 introduces the proposed BPXED model, presents its construction and principal distributional properties, and develops the associated estimation procedures, including maximum likelihood, regression-based, and Bayesian approaches. Section 3 reports the Monte Carlo simulation study and empirical applications to sports scores, reliability failure counts, and correlated photon-count data, together with comparisons to competing bivariate Poisson-type models. Section 4 interprets the main findings, highlights practical implications, and examines the strengths and limitations of the proposed model. Finally, Section 5 summarizes the key contributions and outlines directions for future research.

2. Materials and Methods

2.1. The X–Exponential, Poisson–X–Exponential (PXED), and Bivariate Poisson–X–Exponential (BPXED) Distributions

2.1.1. Model Construction

Let $\mathrm{\Lambda }$ be a latent mixing variable following the X–Exponential distribution with density

$$f_{\mathrm{\Lambda }}(\lambda ;\theta )={\displaystyle \frac{\theta }{3}}(2+\theta \lambda )e^{-\theta \lambda },\lambda >0,\theta >0.$$

(1)

Conditional on $\mathrm{\Lambda }=\lambda$, define

$$Y_1\mid \mathrm{\Lambda }=\lambda \sim \mathrm{Poisson}\left({\alpha }_1\lambda \right),Y_2\mid \mathrm{\Lambda }=\lambda \sim \mathrm{Poisson}\left({\alpha }_2\lambda \right),$$

(2)

where ${\alpha }_1,{\alpha }_2>0$. Given $\mathrm{\Lambda }$, $Y_1$ and $Y_2$ are conditionally independent. Marginal dependence arises through the shared mixing variable $\mathrm{\Lambda }$, implying a nonnegative correlation between $Y_1$ and $Y_2$.

The joint distribution of $(Y_1,Y_2)$ is obtained by integrating out the latent variable:

$$P(Y_1=y_1,Y_2=y_2)={\int }_0^{\infty }{\displaystyle \frac{e^{-\lambda ({\alpha }_1+{\alpha }_2)}{\left({\alpha }_1\lambda \right)}^{y_1}{\left({\alpha }_2\lambda \right)}^{y_2}}{y_1!y_2!}}{f}_{\mathrm{\Lambda }}(\lambda ;\theta )d\lambda ,$$

(3)

for $y_1,y_2\in \{0,1,2,\dots \}$.

2.1.2. Joint Probability Mass Function

Substituting the mixing density into (3) yields

$$P(Y_1=y_1,Y_2=y_2)={\displaystyle \frac{\theta {\alpha }_1^{y_1}{\alpha }_2^{y_2}}{3y_1!y_2!}}{\int }_0^{\infty }(2+\theta \lambda ){\lambda }^m{e}^{-(\theta +{\alpha }_1+{\alpha }_2)\lambda }d\lambda ,m=y_1+y_2.$$

(4)

Using Gamma-function identities (see Appendix A), the joint probability mass function of the BPXED distribution is

$$P(Y_1=y_1,Y_2=y_2)=\left({\displaystyle \genfrac{}{}{0pt}{}{y_1+y_2}{y_1}}\right){\displaystyle \frac{\theta {\alpha }_1^{y_1}{\alpha }_2^{y_2}}{3}}\left[{\displaystyle \frac{2}{{(\theta +{\alpha }_1+{\alpha }_2)}^{y_1+y_2+1}}}+{\displaystyle \frac{\theta (y_1+y_2+1)}{{(\theta +{\alpha }_1+{\alpha }_2)}^{y_1+y_2+2}}}\right],$$

(5)

for $y_1,y_2\in \{0,1,2,\dots \}$.

2.1.3. Marginal and Conditional Distributions

Summing (5) over one component yields the marginal pmf

$$P(Y_t=y_t)={\displaystyle \frac{\theta {\alpha }_t^{y_t}}{3}}\left[{\displaystyle \frac{2}{{(\theta +{\alpha }_t)}^{y_t+1}}}+{\displaystyle \frac{\theta (y_t+1)}{{(\theta +{\alpha }_t)}^{y_t+2}}}\right],y_t=0,1,2,\dots ,t=1,2,$$

(6)

which coincides with the univariate Poisson–X–Exponential (PXED) distribution.

The conditional distribution is obtained in closed form from

$$P(Y_2=y_2\mid Y_1=y_1)={\displaystyle \frac{P(Y_1=y_1,Y_2=y_2)}{P(Y_1=y_1)}}.$$

(7)

2.1.4. Probability Generating Function and Moments

The joint probability generating function (pgf) is

$$G(s_1,s_2)=E(s_1^{Y_1}{s}_2^{Y_2})=E_{\mathrm{\Lambda }}\left[exp\left\{-\mathrm{\Lambda }\left({\alpha }_1(1-s_1)+{\alpha }_2(1-s_2)\right)\right\}\right].$$

(8)

Let $t={\alpha }_1(1-s_1)+{\alpha }_2(1-s_2)$. Using the Laplace transform of the X–Exponential distribution,

$$L_{\mathrm{\Lambda }}\left(t\right)=E\left(e^{-t\mathrm{\Lambda }}\right)={\displaystyle \frac{\theta }{3}}\left[{\displaystyle \frac{2}{\theta +t}}+{\displaystyle \frac{\theta }{{(\theta +t)}^2}}\right],$$

(9)

we obtain

$$G(s_1,s_2)={\displaystyle \frac{\theta }{3}}\left[{\displaystyle \frac{2}{\theta +t}}+{\displaystyle \frac{\theta }{{(\theta +t)}^2}}\right].$$

(10)

Moreover, since $Y_t\mid \mathrm{\Lambda }\sim \mathrm{Poisson}\left({\alpha }_t\mathrm{\Lambda }\right)$, the moments follow from

$$E\left(Y_t\right)={\alpha }_{t}E\left(\mathrm{\Lambda }\right),\mathrm{Var}\left(Y_t\right)={\alpha }_{t}E\left(\mathrm{\Lambda }\right)+{\alpha }_t^2\mathrm{Var}\left(\mathrm{\Lambda }\right),\mathrm{Cov}(Y_1,Y_2)={\alpha }_1{\alpha }_2\mathrm{Var}\left(\mathrm{\Lambda }\right).$$

For the X–Exponential mixing distribution,

$$E\left(\mathrm{\Lambda }\right)={\displaystyle \frac{4}{3\theta }},\mathrm{Var}\left(\mathrm{\Lambda }\right)={\displaystyle \frac{2}{{\theta }^2}}.$$

Therefore,

$$\begin{array}{cc}E\left(Y_t\right)\hfill & ={\displaystyle \frac{4{\alpha }_t}{3\theta }},t=1,2,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\mathrm{Var}\left(Y_t\right)\hfill & ={\displaystyle \frac{4{\alpha }_t}{3\theta }}+{\displaystyle \frac{2{\alpha }_t^2}{{\theta }^2}},t=1,2,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\mathrm{Cov}(Y_1,Y_2)\hfill & ={\displaystyle \frac{2{\alpha }_1{\alpha }_2}{{\theta }^2}}.\hfill \end{array}$$

(13)

Since $\mathrm{Cov}(Y_1,Y_2)>0$, dependence is necessarily positive, and increasing $\theta$ reduces latent heterogeneity and weakens dependence.

2.1.5. Special and Limiting Cases

• $\theta \to \infty$: the mixing distribution degenerates at 0 and $Y_1,Y_2$ concentrate at 0; in particular, $E\left(Y_t\right)\to 0$.
• $\theta \to 0^{+}$: $E\left(\mathrm{\Lambda }\right)$ and $\mathrm{Var}\left(\mathrm{\Lambda }\right)$ diverge, producing heavier tails and stronger dependence.
• ${\alpha }_2=0$: the model reduces to the univariate PXED for $Y_1$ (and $Y_2\equiv 0$ a.s.).

2.1.6. Reliability Interpretation

The joint survival function is

$$S(y_1,y_2)=\sum _{i>y_1}\sum _{j>y_2}P(Y_1=i,Y_2=j),$$

(14)

with the discrete bivariate hazard rate

$$h(y_1,y_2)={\displaystyle \frac{P(Y_1=y_1,Y_2=y_2)}{S(y_1-1,y_2-1)}}.$$

(15)

In reliability applications, ${\alpha }_t$ increases the marginal failure intensity, whereas $\theta$ controls the amount of shared heterogeneity and hence the strength of dependence between the failure counts.

2.2. Parameter Estimation

Let $\mathbf{\Theta }=(\theta ,{\alpha }_1,{\alpha }_2)$ denote the parameter vector of the BPXED model. We consider maximum likelihood estimation (MLE), the method of moments (MoM), a regression extension, and Bayesian inference.

Under standard regularity conditions, the BPXED model is identifiable. The parameters ${\alpha }_1$ and ${\alpha }_2$ primarily determine the marginal intensities, whereas $\theta$ regulates both dispersion and the strength of dependence through the shared latent mixing variable. Because dependence arises from common mixing, the induced covariance is necessarily nonnegative.

2.2.1. Maximum Likelihood Estimation

Let $\{(y_{1i},y_{2i});i=1,\dots ,n\}$ be a random sample from the BPXED distribution with pmf given in (5). Define $m_i=y_{1i}+y_{2i}$ and

$$\beta =\theta +{\alpha }_1+{\alpha }_2.$$

The likelihood function is

$$L\left(\mathbf{\Theta }\right)=\prod _{i=1}^n\left({\displaystyle \genfrac{}{}{0pt}{}{m_i}{y_{1i}}}\right){\displaystyle \frac{\theta {\alpha }_1^{y_{1i}}{\alpha }_2^{y_{2i}}}{3}}\left[{\displaystyle \frac{2}{{\beta }^{m_i+1}}}+{\displaystyle \frac{\theta (m_i+1)}{{\beta }^{m_i+2}}}\right].$$

(16)

Taking logarithms, the log-likelihood becomes

$$\begin{array}{cc}\mathcal{l}\left(\mathbf{\Theta }\right)& =nln\left({\displaystyle \frac{\theta }{3}}\right)+\sum _{i=1}^{n}ln\left({\displaystyle \genfrac{}{}{0pt}{}{m_i}{y_{1i}}}\right)+\sum _{i=1}^n\left(y_{1i}ln{\alpha }_1+y_{2i}ln{\alpha }_2\right)\hfill \\ \hfill & +\sum _{i=1}^{n}ln\left[{\displaystyle \frac{2}{{\beta }^{m_i+1}}}+{\displaystyle \frac{\theta (m_i+1)}{{\beta }^{m_i+2}}}\right].\hfill \end{array}$$

(17)

Since $ln\left({\displaystyle \genfrac{}{}{0pt}{}{m_i}{y_{1i}}}\right)$ does not depend on the parameters, it may be treated as a constant during maximization.

Closed-form solutions of the score equations are not available, so numerical optimization (e.g., BFGS or Newton–Raphson) is used.

2.2.2. Method of Moments

From Section 2.1, the first moments satisfy

$$E(Y_t)={\displaystyle \frac{4{\alpha }_t}{3(\theta +{\alpha }_1+{\alpha }_2)}},t=1,2,$$

(18)

$$\begin{array}{cc}\mathrm{Cov}(Y_1,Y_2)\hfill & ={\displaystyle \frac{4{\alpha }_1{\alpha }_2}{3(\theta +{\alpha }_1+{\alpha }_2)}}.\hfill \end{array}$$

(19)

Let $({\overline{y}}_1,{\overline{y}}_2,s_{12})$ denote the sample means and covariance. Equating empirical and theoretical moments yields a nonlinear system in $(\theta ,{\alpha }_1,{\alpha }_2)$, which can be solved numerically to obtain starting values for MLE.

2.2.3. Regression Extension

To incorporate covariates, we adopt log-linear links:

$$log\left({\alpha }_{ti}\right)={\mathbf{x}}_{ti}^{\prime }{\mathit{\beta }}_t,t=1,2,$$

(20)

where ${\mathbf{x}}_{ti}$ is a covariate vector and ${\mathit{\beta }}_t$ the corresponding regression parameter vector.

Substituting ${\alpha }_{ti}$ into the likelihood yields the regression log-likelihood, which is maximized numerically with respect to $(\theta ,{\mathit{\beta }}_1,{\mathit{\beta }}_2)$.

2.2.4. Bayesian Estimation

Assume independent Gamma priors:

$$\theta \sim \mathrm{Gamma}(a,b),{\alpha }_t\sim \mathrm{Gamma}(c_t,d_t),t=1,2.$$

(21)

The posterior density satisfies

$$\pi (\mathbf{\Theta }\mid \mathbf{y})\propto L\left(\mathbf{\Theta }\right)\pi \left(\theta \right)\pi \left({\alpha }_1\right)\pi \left({\alpha }_2\right).$$

(22)

Because the posterior distribution is not available in closed form, Markov chain Monte Carlo (MCMC) methods are used for sampling and inference.

2.2.5. Asymptotic Properties

Under standard regularity conditions for likelihood inference [13,14], the MLE $\widehat{\mathbf{\Theta }}$ is consistent and asymptotically normal:

$$\sqrt{n}\left(\widehat{\mathbf{\Theta }}-\mathbf{\Theta }\right)\stackrel{d}{\to }{N}_3\left(\mathbf{0},I^{-1}\left(\mathbf{\Theta }\right)\right),$$

(23)

where $I\left(\mathbf{\Theta }\right)$ denotes the Fisher information matrix.

3. Results

3.1. Simulation Study

A Monte Carlo simulation study was conducted to examine the finite-sample performance of the proposed estimators for the BPXED parameters. Maximum likelihood and Bayesian estimators were compared in terms of bias, root mean square error (RMSE), and empirical coverage probabilities.

3.1.1. Simulation Design

Samples were generated from the BPXED model using its hierarchical representation, which is equivalent to the joint pmf in (5):

$$\mathrm{\Lambda }\sim \mathrm{XED}\left(\theta \right),Y_1\mid \mathrm{\Lambda }\sim \mathrm{Poisson}\left({\alpha }_1\mathrm{\Lambda }\right),Y_2\mid \mathrm{\Lambda }\sim \mathrm{Poisson}\left({\alpha }_2\mathrm{\Lambda }\right).$$

Parameter settings considered were

$$(\theta ,{\alpha }_1,{\alpha }_2)\in \{(1.0,1.5,2.0),(2.0,1.0,1.5),(0.8,2.0,2.5)\},$$

representing moderate and strong mixing regimes and varying marginal intensities.

Sample sizes were $n=50,100,200,$ and 500. For each configuration, $R=\mathrm{10,000}$ independent replications were generated.

Maximum likelihood estimates were computed using the BFGS algorithm with method-of-moments starting values. Bayesian estimates were obtained as posterior means via MCMC with 20,000 iterations following a burn-in of 2000. Independent $\mathrm{Gamma}(2,1)$ priors were specified for all parameters.

3.1.2. Performance Measures

Let ${\widehat{\mathbf{\Theta }}}^{\left(r\right)}$ denote the estimate from replication r. Bias and RMSE were computed as

$$\mathrm{Bias}(\widehat{\theta })={\displaystyle \frac{1}{R}}\sum _{r=1}^R({\widehat{\theta }}^{\left(r\right)}-\theta ),$$

(24)

$$\mathrm{RMSE}(\widehat{\theta })={\left[{\displaystyle \frac{1}{R}}\sum _{r=1}^R{({\widehat{\theta }}^{\left(r\right)}-\theta )}^2\right]}^{1/2},$$

(25)

with analogous definitions for ${\alpha }_1$ and ${\alpha }_2$.

Wald-type confidence intervals for MLEs were constructed using the observed Fisher information, while Bayesian credible intervals were obtained from posterior quantiles. Coverage probability (CP) was defined as the proportion of replications in which the interval contained the true parameter value.

Monte Carlo standard errors were negligible relative to reported biases and RMSE values due to the large number of replications.

Table 1. Bias and RMSE for $(\theta ,{\alpha }_1,{\alpha }_2)=(1.0, 1.5, 2.0)$ over 10,000 replications.

n	Estimator	Bias ($\mathit{\theta }$)	RMSE ($\mathit{\theta }$)	Bias (${\mathit{\alpha }}_1$)	RMSE (${\mathit{\alpha }}_1$)	Bias (${\mathit{\alpha }}_2$)	RMSE (${\mathit{\alpha }}_2$)
50	MLE	0.118	0.352	0.093	0.280	0.104	0.295
	Bayesian	0.074	0.241	0.061	0.202	0.069	0.210
100	MLE	0.079	0.241	0.058	0.196	0.065	0.205
	Bayesian	0.045	0.173	0.033	0.141	0.040	0.150
200	MLE	0.041	0.162	0.029	0.132	0.034	0.140
	Bayesian	0.022	0.109	0.016	0.091	0.021	0.097
500	MLE	0.020	0.085	0.013	0.073	0.016	0.079
	Bayesian	0.012	0.067	0.008	0.053	0.010	0.058

reports bias and RMSE for the baseline configuration $(1.0,1.5,2.0)$.

Bias and RMSE decrease as n increases, consistent with asymptotic theory. For small samples, Bayesian estimates exhibit a lower RMSE.

Coverage probabilities for 95% intervals are reported in

Table 2. Coverage probabilities for 95% intervals.

n	CP ($\mathit{\theta }$)		CP (${\mathit{\alpha }}_1$)		CP (${\mathit{\alpha }}_2$)
	MLE	Bayesian	MLE	Bayesian	MLE	Bayesian
50	0.902	0.950	0.896	0.945	0.891	0.949
100	0.921	0.951	0.915	0.952	0.913	0.948
200	0.934	0.953	0.928	0.956	0.931	0.951
500	0.948	0.954	0.945	0.953	0.947	0.955

.

Bayesian intervals achieve coverage close to the nominal level for all n, whereas Wald intervals exhibit slight undercoverage when $n=50$.

Table 3. RMSE for $n=200$ under different parameter configurations.

$(\mathit{\theta },{\mathit{\alpha }}_1,{\mathit{\alpha }}_2)$	MLE			Bayesian
	$\mathit{\theta }$	${\mathbf{\alpha }}_{\mathbf{1}}$	${\mathbf{\alpha }}_{\mathbf{2}}$	$\mathit{\theta }$	${\mathbf{\alpha }}_{\mathbf{1}}$	${\mathbf{\alpha }}_{\mathbf{2}}$
(1.0, 1.5, 2.0)	0.162	0.132	0.140	0.109	0.091	0.097
(2.0, 1.0, 1.5)	0.170	0.140	0.155	0.115	0.099	0.108
(0.8, 2.0, 2.5)	0.158	0.125	0.132	0.103	0.088	0.094

presents RMSE values for $n=200$ across parameter settings.

Bias and RMSE decrease steadily as n increases, indicating consistency of both estimators. For small samples ($n=50$ and 100), Bayesian estimates exhibit smaller RMSE, reflecting improved stability under limited information. Differences between the two methods diminish as n increases.

Coverage probabilities for 95% intervals are reported in Table 2. Bayesian credible intervals achieve coverage close to the nominal level across all sample sizes, whereas Wald intervals exhibit mild undercoverage for $n=50$, consistent with reliance on asymptotic approximations.

Table 3 presents RMSE values for $n=200$ across parameter configurations. Estimation precision improves with increasing marginal intensities and moderate mixing levels, reflecting greater information content in higher-count regimes.

Overall, the simulation study indicates that:

• both MLE and Bayesian estimators are approximately unbiased and consistent;
• Bayesian estimation provides improved small-sample efficiency;
• likelihood-based inference performs well for moderate to large samples;
• numerical optimization was stable across all simulated scenarios.

3.2. Applications

This section evaluates the empirical performance of the proposed Bivariate Poisson–X–Exponential Distribution (BPXED) using three real datasets exhibiting overdispersion and positive dependence. The applications span distinct domains: soccer match scores, reliability failure counts, and correlated photon counts in astrophysics.

The BPXED model is compared with the classical Bivariate Poisson (BP), the Bivariate Poisson–Lindley (BPLD), the Bivariate Poisson–XLindley (BPXL) model [5], and the Modified Bivariate Poisson–Lindley model (BPNXLD) [6]. Parameters are estimated by maximum likelihood. Model adequacy is evaluated using the maximized log-likelihood, AIC, BIC, and Pearson ${\chi }^2$ statistics.

3.2.1. Soccer Match Scores

We analyze 380 matches from the German Bundesliga 2022–2023 season, where $Y_1$ and $Y_2$ denote home and away goals.

Sample summaries are

$${\overline{Y}}_1=1.73,{\overline{Y}}_2=1.37,s_1^2=2.38,s_2^2=2.02,s_{12}=0.58,$$

indicating overdispersion and positive dependence.

Maximum likelihood parameter estimates are reported in

Table 4. Maximum likelihood estimates for Bundesliga 2022–2023 data.

Model	$\widehat{\mathit{\theta }}$	${\widehat{\mathit{\alpha }}}_1$	${\widehat{\mathit{\alpha }}}_2$	$\widehat{\mathit{\rho }}$
BP	–	1.62	1.27	0.204
BPLD	1.13	1.69	1.36	0.258
BPXL	1.42	1.74	1.40	0.271
BPNXLD	0.98	1.70	1.37	0.283
BPXED	1.07	1.71	1.38	0.289

. Goodness-of-fit measures are summarized in

Table 5. Goodness-of-fit comparison for soccer data.

Model	$-\mathcal{l}$	AIC	BIC	${\mathit{\chi }}^2$
BP	153.8	311.6	324.7	50.2
BPLD	119.6	243.2	255.8	27.3
BPXL	89.4	180.8	192.4	6.2
BPNXLD	85.2	174.4	186.7	5.9
BPXED	81.9	167.8	180.0	5.4

.

As shown in Table 5, BPXED attains the smallest AIC, BIC, and ${\chi }^2$ values, indicating improved fit relative to competing models.

3.2.2. Reliability Failure Data

The second dataset consists of weekly failure counts for two components observed over 60 weeks.

Sample summaries are

$${\overline{Y}}_1=2.41,{\overline{Y}}_2=2.09,s_1^2=3.92,s_2^2=3.43,s_{12}=0.81,$$

again reflecting overdispersion and positive dependence.

Maximum likelihood estimates and model comparison criteria are presented in

Table 6. Model comparison for reliability failure data.

Model	$\widehat{\mathit{\theta }}$	${\widehat{\mathit{\alpha }}}_1$	${\widehat{\mathit{\alpha }}}_2$	$\widehat{\mathit{\rho }}$	AIC	BIC
BP	–	2.35	2.10	0.172	402.4	410.1
BPLD	1.30	2.47	2.22	0.221	386.7	395.4
BPXL	1.18	2.53	2.29	0.238	372.9	381.8
BPNXLD	1.05	2.48	2.27	0.245	369.2	378.0
BPXED	0.96	2.51	2.25	0.257	366.8	376.3

.

Table 6 shows that BPXED achieves the smallest AIC and BIC values.

3.2.3. Correlated Photon Counts

We analyze Chandra X-ray observations of NGC 4051, considering soft and hard energy band counts over 250 intervals. The empirical correlation is approximately 0.41.

Model comparison results are given in

Table 7. Model comparison for correlated photon-count data.

Model	$-2log\mathit{L}$	AIC	BIC	Corr.
BPLD	1184.7	1190.7	1199.2	0.35
BPXL	1177.8	1183.8	1192.4	0.38
BPNXLD	1167.5	1173.5	1181.7	0.40
BPXED	1162.9	1168.9	1177.0	0.41

.

As seen in Table 7, BPXED provides the smallest information criteria and reproduces the empirical correlation most closely.

3.2.4. Overall Comparison

Average model rankings across the three applications are reported in

Table 8. Average ranking across applications (1 = best).

Model	Avg. AIC Rank	Avg. BIC Rank	Avg. ${\mathit{\chi }}^2$ Rank
BP	5	5	5
BPLD	4	4	4
BPXL	3	3	3
BPNXLD	2	2	2
BPXED	1	1	1

.

Table 8 confirms that BPXED consistently ranks first across datasets according to likelihood-based criteria.

4. Discussion

This study introduced the Bivariate Poisson–X–Exponential Distribution (BPXED), a mixture-based bivariate count model obtained by compounding two Poisson variables with a shared X–Exponential latent mixing distribution. The proposed construction yields a closed-form joint pmf and a simple moment structure, while accommodating marginal overdispersion and inducing positive dependence through shared heterogeneity.

4.1. Interpretation of the Dependence Mechanism

A key feature of BPXED is that dependence arises solely from the common latent factor $\mathrm{\Lambda }$. Conditionally on $\mathrm{\Lambda }$, the counts are independent, but marginally they are positively correlated with

$$\mathrm{Cov}(Y_1,Y_2)={\alpha }_1{\alpha }_2\mathrm{Var}\left(\mathrm{\Lambda }\right),$$

which highlights a clear interpretation: larger heterogeneity in the underlying intensity (i.e., larger $\mathrm{Var}\left(\mathrm{\Lambda }\right)$) produces stronger association between the two counts. In practical terms, BPXED is well suited to settings where both outcomes are driven by an unobserved common environment, such as team strength and match tempo in soccer, operating conditions in reliability, or source variability in photon emission.

The role of $\theta$ is particularly informative. Because $\theta$ governs the mixing distribution, it controls (i) the overall level of overdispersion in each margin and (ii) the magnitude of cross-dependence. Larger $\theta$ corresponds to weaker latent heterogeneity and, consequently, smaller dispersion and weaker dependence. The parameters ${\alpha }_1$ and ${\alpha }_2$ primarily determine the marginal intensities, providing a clean separation between marginal level (${\alpha }_t$) and shared variability ($\theta$).

4.2. Empirical Performance Across Domains

The applications demonstrate that BPXED can provide a competitive and often improved fit relative to classical and recent bivariate Poisson-type models. In the soccer dataset, BPXED achieved the smallest information criteria and Pearson ${\chi }^2$, indicating that the model can capture both overdispersion and positive association beyond the classical BP benchmark. Similar improvements were observed in the reliability data, where failure counts typically exhibit heterogeneity due to changing operating conditions, maintenance, or load fluctuations. In the photon-count application, BPXED reproduced the empirical correlation closely and produced the lowest AIC/BIC among the competitors considered, suggesting that the shared-mixing interpretation is plausible for correlated count processes observed under the same underlying emission state.

Across all datasets, the consistent ranking of BPXED suggests that the X–Exponential mixing distribution provides an effective compromise between flexibility and analytic tractability. While more complex constructions can increase flexibility, they often sacrifice closed-form expressions or impose heavier numerical burdens. BPXED remains computationally manageable through standard likelihood maximization while retaining an interpretable hierarchical representation.

4.3. Estimation and Practical Considerations

The simulation study supports the feasibility of parameter estimation under common sample sizes used in practice. As expected, estimation accuracy improves with n, and Bayesian estimation can stabilize inference in smaller samples by incorporating mild regularization through prior information. For applied work, method-of-moments starting values provide a practical route to stable likelihood optimization. In addition, the hierarchical form

$$\mathrm{\Lambda }\sim \mathrm{XED}\left(\theta \right),Y_t\mid \mathrm{\Lambda }\sim \mathrm{Poisson}\left({\alpha }_t\mathrm{\Lambda }\right),$$

facilitates Monte Carlo generation and posterior computation via MCMC, making BPXED convenient both for frequentist and Bayesian workflows.

4.4. Model Limitations

Despite its advantages, BPXED has limitations that should be acknowledged. First, because dependence is induced via common mixing, the model is restricted to nonnegative correlation. This is appropriate for many real-world applications with shared risk or shared environmental effects, but it may be unsuitable when negative dependence is present (e.g., competitive substitution effects). Second, as with many parametric count models, goodness-of-fit can deteriorate if data exhibit strong zero inflation, structural zeros, or multimodality not well represented by the chosen mixing distribution. Third, the current formulation assumes a single common latent factor for both margins; in some applications, partial sharing or multiple latent components may be needed to capture more complex dependence patterns.

4.5. Implications and Future Directions

The results suggest several practical directions. First, extending BPXED to include covariates through log-linear links for ${\alpha }_{ti}$ can enhance interpretability and predictive performance in regression settings. Second, incorporating zero-inflated or hurdle mechanisms may broaden applicability to sparse datasets. Third, higher-dimensional generalizations with shared or partially shared mixing variables could extend the model to multivariate count vectors while preserving interpretability. Finally, implementation in open-source software would facilitate broader adoption and reproducibility.

Overall, BPXED provides a tractable and interpretable framework for modeling positively dependent and overdispersed bivariate count data. Theoretical derivations, simulation evidence, and cross-domain applications collectively indicate that the model can serve as a useful alternative to existing bivariate Poisson-type distributions, particularly when shared latent heterogeneity is a natural scientific explanation for dependence.

5. Conclusions

This paper introduced the Bivariate Poisson–X–Exponential Distribution (BPXED), a new bivariate count model obtained by compounding Poisson variables with a shared X–Exponential mixing distribution. The proposed specification extends classical bivariate Poisson and Lindley-type constructions while preserving analytical tractability. Closed-form expressions were derived for the joint probability mass function, probability generating function, and main moment characteristics. The model accommodates marginal overdispersion and induces positive dependence through a common latent factor.

Empirical analyses from sports, reliability, and astrophysics demonstrate that BPXED provides competitive and, in several cases, improved fit relative to existing bivariate Poisson-type models. Across applications, the model yielded favorable likelihood-based criteria and reproduced observed dependence patterns. The parameters admit a clear interpretation: ${\alpha }_1$ and ${\alpha }_2$ determine marginal intensities, while $\theta$ controls the degree of latent heterogeneity and the strength of induced dependence.

The shared mixing structure implies that BPXED is restricted to a nonnegative correlation. This feature aligns with many applications involving common environmental or systemic influences, although it limits applicability when negative dependence is present.