For many years there has been an increase in the statistical research to develop multivariate distributions for count data. There are numerous ways by which we can generalize the distributions of univariate counts to multivariate forms. The applications of these multivariate distributions are found in many fields such as engineering, bioinformatics, genetic epidemiology and biomedicine. An interested reader is referred to the recent editions of the book by Winkelmann [1] and the book by Cameron and Trivedi [2] for thorough reviews of regression models for multivariate count data and the very recent work by Tzougas et al. [3] . Needless to say, fitting these multivariate distribution to data is quite complicated and requires a great deal of computing resources. Therefor there is a need to develop more flexible bivariate models and demonstrate their applicability to real life data.

The main purpose of this article is to investigate the statistical characteristics of a class of bivariate generalized Poisson models with a correlation parameter that offers sufficient flexibility for accommodating overdispersion and accounting for the positive correlation between the variables of interest. The distribution is named “Bivariate Generalized Poisson Distribution”, BGPD, which is a special member of a larger family of distributions developed by Shenton and Consul [4] . Some of the interesting properties of this class were studies further by Jain and Sing [5] , and by Shoukri [6] who investigated some of the properties of the distribution as a member of the class of bivariate modified power series distributions.

This paper has three-fold objectives. Firstly, we derive the locally powerful score test on the hypothesis that the correlation parameter is significantly different from zero, and then assess the power of this test. Secondly, construct confidence interval on the ratio of the means of the two correlated Poisson variables. Thirdly, we fit the model to metabolic syndrome data available from a random sample of spousal pairs.

A bivariate random variable (X, Y) is said to have BGPD when the joint probability distribution is given by (1):

$\begin{array}{c}P\left(X=x,Y=y\right)=\frac{{\left(1+m_{1}x+m_{2}y\right)}^{x+y-1}}{x!y!}{\theta }_1^x{\theta }_2^y\exp [-\left({\theta }_1+{\theta }_2\right)\\ \cdot \left(1+m_{1}x+m_{2}y\right)]\end{array}$(1)

$x,y\succcurlyeq 0$, $0<{\theta }_i<\infty$, $m_i>0\left(i=1,2\right)$, $0<{\theta }_1{m}_1+{\theta }_2{m}_2<1$, and zero otherwise. The BGPD can be obtained as follows:

By expanding the bivariate probability generating function (BPGF) $\varphi \left(t_1,t_2\right)=\exp \left[{\theta }_1\left(t_1-1\right)+{\theta }_2\left(t_2-1\right)\right]$, using the bivariate Lagrange expansion of an implicit function, as was shown in [4] .

From (1) we can write

$\begin{array}{c}\exp \left({\theta }_1+{\theta }_2\right)=\underset{x,y}{{\displaystyle \sum }}{\left(x!y!\right)}^{-1}{\left(1+m_{1}x+m_{2}y\right)}^{x+y-1}\\ \cdot {\theta }_1^x{\theta }_2^y\exp \left[-\left({\theta }_1+{\theta }_2\right)\left(1+m_{1}x+m_{2}y\right)\right]\end{array}$(2)

Therefore, from (2) we can see that the distribution posses a power series expansion in terms of the parameters ( ${\theta }_1$ & ${\theta }_2$).

On differentiating the above equation {2} partially w.r.t. ${\theta }_1$ and ${\theta }_2$ respectively, dividing by $\exp \left({\theta }_1+{\theta }_2\right)$, and on summation, we get by solving for ${\mu }_{1,0}$ and ${\mu }_{0,1}$.

${\mu }_{1,0}={\theta }_1{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-1}$(3)

${\mu }_{0,1}={\theta }_2{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-1}$(4)

To obtain a recurrence relation among the higher non-central product moments, we write

${\mu }_{r\cdot s}=\underset{x,y}{{\displaystyle \sum }}{x}^r{y}^{s}P\left(X=x,Y=y\right)$(5)

Define the cross product moments:

${\mu }_{r+1,s}=E\left[{\left(x-{{\mu }^{\prime }}_{10}\right)}^{r+1}{\left(y-{{\mu }^{\prime }}_{01}\right)}^s\right]$.

We can show that

${\mu }_{r+1,s}={\theta }_1{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-1}\left[\left\{\left(1-m_2{\theta }_2\right){\partial }_1+m_2{\theta }_2{\partial }_2\right\}{{\mu }^{\prime }}_{r,s}\right]+{{\mu }^{\prime }}_{r,s}{{\mu }^{\prime }}_{1,0}$(6)

By symmetry we can write:

${{\mu }^{\prime }}_{r,s+1}={\theta }_2{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-1}\left[\left\{\left(1-m_1{\theta }_1\right){\partial }_2+m_1{\theta }_1{\partial }_1\right\}{{\mu }^{\prime }}_{r,s}\right]+{{\mu }^{\prime }}_{r,s}{{\mu }^{\prime }}_{0,1}$(7)

These relationships among the moments are obtained on following Kendall [7] .

In particular, from (6) and (7):

${\mu }_{1,0}={\theta }_1{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-1}$(8)

${\mu }_{2,0}={\theta }_1{\left(1-m_1{\theta }_1-m_2{\theta }_2\right)}^{-3}\left[{\left(1-m_2{\theta }_2\right)}^2+{\theta }_1{\theta }_2{m}_2^2\right]$(9)

are respectively the mean and variance of X in terms of the model parameters.

And the values of ${\mu }_{0,1}$ and ${\mu }_{0,2}$ can be written down by symmetry. The coefficient of correlation $\rho$, between X and Y is given by

$\rho =\frac{{\theta }_1{\theta }_2\left[m_1\left(1-m_2{\theta }_2\right)+m_2\left(1-m_1{\theta }_1\right)\right]}{{\left[{\theta }_1{\theta }_2\left(1-2m_1{\theta }_1+m_1^2{\theta }_1^2+{\theta }_1{\theta }_2{m}_1^2\right)\left(1-2m_2{\theta }_2+m_2^2{\theta }_2^2+{\theta }_1{\theta }_2{m}_2^2\right)\right]}^{1/2}}$(10)

As can be seen, from the restrictions on the model parameters, the correlation (10) cannot be negative.

The metabolic syndrome is the co-aggregation of hypertension, impaired glucose tolerance, dyslipidemia, and abdominal obesity and is associated with an increased risk of total and cardiovascular mortality in adults [9] [10] . Genetics as well as environmental influences have been implicated in obesity and several cardiovascular risk factors [11] [12] . Family is one of the most important factors affecting metabolic risk factors in children, in that family displays an interaction between genetic and shared environmental factors [13] [14] . Recent research showed that childhood and adolescent overweight has been increasing in Asian countries due to urbanization and economic development. For example, over the past 10 years, the rates of overweight among Korean children and adolescents aged 5 - 20 years have doubled [15] , which may ultimately cause an increase in adverse cardiovascular outcomes. Globally, the prevalence of the metabolic syndrome is high among obese children and adolescents, and increases with increasing obesity [16] . The widely used definition of metabolic syndrome is that of the World Health Organization [17] [18] . The components of each definition and criteria for making the diagnosis of the metabolic syndrome are summarized in Table 1 .

The abbreviations of the medical official bodies given in Table 1 are:

IDF ≡ International Diabetes Federation, WHO ≡ World Health Organization, EGIR ≡ European Group for the Study of Insulin Resistance, and NCEP ≡ National Cholesterol Education Program.

The Metabolic syndrome data:

Based on a random sample of 4000 spousal pairs, the female and male counts are cross classified in Table 2 and Table 3 .

The test independence had a value of U= 7.027, with a corresponding p-value= 2.11e−12.

Therefore the hypothesis of independence is not supported by the metabolic

syndrome data.

To fit the model to the data we shall use the method of moments to estimate the three population parameters. We shall use the notations ${\mu }_{10}^{*}$ & ${\mu }_{01}^{*}$ to respectively denotes the sample means of x & y, while ${\mu }_{11}^{*}$ denotes the sample covariance between x & y.

${\theta }_1^{*}=\frac{{\mu }_{10}^{*}}{1+m^{*}\left({\mu }_{10}^{*}+{\mu }_{10}^{*}\right)}$ (29)

${\theta }_2^{*}=\frac{{\mu }_{01}^{*}}{1+m^{*}\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right)}$ (30)

$m^{*2}\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right){\mu }_{10}^{*}{\mu }_{01}^{*}+2m^{*}{\mu }_{10}^{*}{\mu }_{01}^{*}-{\mu }_{11}^{*}=0$ (31)

From Equations (29)-(31), the admissible moment estimator for $m$ is.

$m^{*}=\frac{{\left[{\left({\mu }_{10}^{*}{\mu }_{01}^{*}\right)}^2+\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right){\mu }_{11}^{*}{\mu }_{10}^{*}{\mu }_{01}^{*}\right]}^{1/2}-{\mu }_{10}^{*}{\mu }_{01}^{*}}{\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right){\mu }_{10}^{*}{\mu }_{01}^{*}}$ (32)

The moments estimators of ${\theta }_1$ & ${\theta }_2$ are given respectively by ${\theta }_1^{*}$ & ${\theta }_2^{*}$ in Equations (33) and (34).

${\theta }_1^{*}=\frac{{\mu }_{10}^{*2}{\mu }_{01}^{*}}{{\left[{\left({\mu }_{10}^{*}{\mu }_{01}^{*}\right)}^2+\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right){\mu }_{11}^{*}{\mu }_{10}^{*}{\mu }_{01}^{*}\right]}^{1/2}}$ (33)

${\theta }_2^{*}=\frac{{\mu }_{01}^{*2}{\mu }_{10}^{*}}{{\left[{\left({\mu }_{10}^{*}{\mu }_{01}^{*}\right)}^2+\left({\mu }_{10}^{*}+{\mu }_{01}^{*}\right){\mu }_{11}^{*}{\mu }_{10}^{*}{\mu }_{01}^{*}\right]}^{1/2}}$ (34)

From the metabolic syndrome data we have, for females, ${\theta }_1^{*}=0.294$, and for male, ${\theta }_2^{*}=0.074$, and $m^{*}=1.262$. The familiar chi-square goodness of fit, with 4 degrees of freedom had a p-value = 0.141. This shows that the model gives a reasonable fit to the data.

The following is a short R code from which we evaluate the nonparametric confidence interval and assess the large sample distribution of the ratio of correlated Poisson means.

R-CODE for bootstrapping the confidence interval on the ratio of means

x<-data$Male_count

y<-data$Female_count

df<-data.frame(x,y)

library(boot)

boot_df=boot(df,function(i,d)mean(i[d] [1] )/mean(i[d] [2] ),R=999)

boot_df

Bootstrap Statistics :

original bias std. error

ratio 3.964 0.0163 0.148

It is clear from Figure 1 that the quantile plot of the sample ratio of means follows a normal distribution ( Table 4 ).

In this article, we discussed some of the interesting applications of a bivariate generalized Poisson distribution. The model has three parameters and has provided satisfactory fit to the metabolic syndrome data collected from spousal pairs. The common parameter (m) captures overdispersion and accurately accounts for the strength of the positive correlation between the two variables.

It is worth noting that this family of models, which is a member of the Lagrange family of bivariate distributions, is suitable for many applications, and in particular in situation where queuing theory is applicable. For example, in large tertiary hospital outpatients departments, the patients usually wait for admission form queues. If we have two types of queues (e.g. males and females), then the random variables would represent the numbers of admitted patients during a busy period of the admission office (single server queue). We also note that the marginal probability distributions have been applied to a variety of data from genomics, and case fatality due to COVID [19] .

Finally, it is worth noting that with the not so complicated structure of the model, it can be shown that it belongs to the class of bivariate of generalized linear models where inclusion of covariates will make this model attractive alternative in many applications.

On using the recurrence relationship in (6) we can show that: Denote

$\Delta =\left(1-m{\theta }_1-m{\theta }_2\right)$

${\mu }_{11}=\frac{m{\theta }_1{\theta }_2\left(1-m{\theta }_1-m{\theta }_2\right)}{{\left(1-m{\theta }_1-m{\theta }_2\right)}^3}$

$\begin{array}{c}{\mu }_{12}=\frac{1}{{\Delta }^4}[m{\theta }_1{\theta }_2^2\left(1-m{\theta }_2\right)-m{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)\left(1-m{\theta }_2\right)\\ +m{\theta }_1{\theta }_2{\left(1-m{\theta }_1\right)}^2+2m^3{\theta }_1^2{\theta }_2^2]\\ +\frac{1}{{\Delta }^5}[3m^4{\theta }_1^2{\theta }_2^3+3m^2{\theta }_1{\theta }_2^2{\left(1-m{\theta }_1\right)}^2+3m^3{\theta }_1^2{\theta }_2^2\left(1-m{\theta }_2\right)\\ +3m{\theta }_1{\theta }_2{\left(1-m{\theta }_1\right)}^2\left(1-m{\theta }_2\right)]\end{array}$

$\begin{array}{c}{\mu }_{22}=E\left[{\left(x-{{\mu }^{\prime }}_{10}\right)}^2{\left(y-{{\mu }^{\prime }}_{01}\right)}^2\right]\\ =\frac{{\theta }_1}{1-m{\theta }_1-m{\theta }_2}\{\left[\left(1-m{\theta }_2\right)\frac{\partial {\mu }_{12}}{\partial {\theta }_1}+m{\theta }_2\frac{\partial {\mu }_{12}}{\partial {\theta }_2}\right]\\ +{\mu }_{02}\left[\frac{{\left(1-m{\theta }_2\right)}^2}{{\left(1-m{\theta }_1-m{\theta }_2\right)}^2}+\frac{m^2{\theta }_1{\theta }_2}{{\left(1-m{\theta }_1-m{\theta }_2\right)}^2}\right]\\ +2{\mu }_{11}\left[\frac{m{\theta }_2\left(1-m{\theta }_2\right)}{{\left(1-m{\theta }_1-m{\theta }_2\right)}^2}+\frac{m{\theta }_2\left(1-m{\theta }_1\right)}{{\left(1-m{\theta }_1-m{\theta }_2\right)}^2}\right]\}\end{array}$

where

$\begin{array}{c}\frac{\partial {\mu }_{12}}{\partial {\theta }_1}=\frac{1}{{\Delta }^4}[m{\theta }_2^2\left(1-m{\theta }_2\right)-m{\theta }_2\left(1-m{\theta }_2\right)\left(1-2m{\theta }_1\right)\\ +m{\theta }_2\left[{\left(1-m{\theta }_1\right)}^2-2m^2{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)+4m^3{\theta }_1{\theta }_2^2\right]\\ +\frac{1}{{\Delta }^5}[4m^2{\theta }_1^2{\theta }_2\left(1-m{\theta }_2\right)-4m^2{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)\times \left(1-m{\theta }_2\right)\\ +4m^2{\theta }_1{\theta }_2{\left(1-m{\theta }_1\right)}^2+8m^4{\theta }_1^2{\theta }_2^2+6m^4{\theta }_1{\theta }_2^3+3m^2{\theta }_2^2{\left(1-m{\theta }_1\right)}^2\\ -6m^3{\theta }_1{\theta }_2^2\left(1-m{\theta }_1\right)+6m^3{\theta }_1{\theta }_2^2\left(1-m{\theta }_2\right)+3m{\theta }_2{\left(1-m{\theta }_1\right)}^2\left(1-m{\theta }_2\right)\\ -6m^2{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)\left(1-m{\theta }_2\right)]+\frac{5m}{{\Delta }^6}[3m^4{\theta }_1^2{\theta }_2^3+3m^2{\theta }_1{\theta }_2^2{\left(1-m{\theta }_1\right)}^2\\ +3m^3{\theta }_1^2{\theta }_2^2\left(1-m{\theta }_2\right)+3m{\theta }_1{\theta }_2{\left(1-m{\theta }_1\right)}^2\left(1-m{\theta }_2\right)]\end{array}$

and

$\begin{array}{c}\frac{\partial {\mu }_{12}}{\partial {\theta }_2}=\frac{1}{{\Delta }^4}[2m{\theta }_1{\theta }_2\left(1-m{\theta }_2\right)-m^2{\theta }_1{\theta }_2^2-m{\theta }_1\left(1-m{\theta }_1\right)\left(1-m{\theta }_2\right)\\ +m^2{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)+m{\theta }_1{\left(1-m{\theta }_1\right)}^2+4m^3{\theta }_1^2{\theta }_2+4m^3{\theta }_1{\theta }_2^2]\\ +\frac{1}{{\Delta }^5}[6m^4{\theta }_1{\theta }_2^3+3m^2{\theta }_2^2{\left(1-m{\theta }_1\right)}^2-6m^3{\theta }_2^2{\theta }_1\left(1-m{\theta }_1\right)\\ +6m^3{\theta }_1{\theta }_2^2\left(1-m{\theta }_2\right)+3m{\theta }_2{\left(1-m{\theta }_1\right)}^2\left(1-m{\theta }_2\right)\end{array}$

$\begin{array}{c}-6m^2{\theta }_1{\theta }_2\left(1-m{\theta }_1\right)\left(1-m{\theta }_2\right)]+\frac{5m}{{\Delta }^6}[3m^4{\theta }_1^2{\theta }_2^3+3m^2{\theta }_1{\theta }_2^2{\left(1-m{\theta }_1\right)}^2\\ +3m^2{\theta }_1^2{\theta }_2^2\left(1-m{\theta }_2\right)+3m{\theta }_1{\theta }_2{\left(1-m{\theta }_1\right)}^2\left(1-m{\theta }_2\right)]\end{array}$

${\mu }_{30}=\frac{m^3{\theta }_1^2{\theta }_2+{\theta }_1{\left(1-m{\theta }_2\right)}^3}{{\Delta }^4}+\frac{3m^3{\theta }_1^3{\theta }_2+3m{\theta }_1^2{\left(1-m{\theta }_2\right)}^2}{{\Delta }^5}$

And from the recurrence relation, we have:

${\mu }_{40}=\frac{{\theta }_1\left(1-m{\theta }_2\right)}{\Delta }\cdot \frac{\partial {\mu }_{30}}{\partial {\theta }_1}+\frac{m{\theta }_1{\theta }_2}{\Delta }\cdot \frac{\partial {\mu }_{30}}{\partial {\theta }_2}+3{\mu }_{20}\left[\frac{{\left(1-m{\theta }_2\right)}^2}{{\Delta }^2}+\frac{m^2{\theta }_1{\theta }_2}{{\Delta }^2}\right]$

The ${\mu }_{40}$ depends on the:

$\frac{\partial {\mu }_{30}}{\partial {\theta }_1}$ & $\frac{\partial {\mu }_{30}}{\partial {\theta }_2}$

We can show that:

$\begin{array}{c}\frac{\partial {\mu }_{30}}{\partial {\theta }_1}=\frac{1}{{\Delta }^4}\left[2m^2{\theta }_1{\theta }_2+{\left(1-m{\theta }_2\right)}^3\right]\\ +\frac{1}{{\Delta }^5}\left[13m^3{\theta }_1^2{\theta }_2+6m{\theta }_1{\left(1-m{\theta }_2\right)}^2+4m{\theta }_1{\left(1-m{\theta }_2\right)}^3\right]\\ +\frac{1}{{\Delta }^6}\left[[15m^4{\theta }_1^3{\theta }_2+15m^2{\theta }_1^2{\left(1-m{\theta }_2\right)}^2\right]\end{array}$

$\begin{array}{c}\frac{\partial {\mu }_{30}}{\partial {\theta }_2}=\frac{1}{{\Delta }^4}\left[m^2{\theta }_1^2-3m{\theta }_1{\left(1-m{\theta }_2\right)}^2\right]\\ +\frac{1}{{\Delta }^5}\left[3m^3{\theta }_1^3-6m^2{\theta }_1^2\left(1-m{\theta }_2\right)+4m^3{\theta }_1^2{\theta }_2+4m{\theta }_1{\left(1-m{\theta }_2\right)}^3\right]\\ +\frac{1}{{\Delta }^6}\left[15m^4{\theta }_1^3{\theta }_2+15m^2{\theta }_1^2{\left(1-m{\theta }_2\right)}^2\right]\end{array}$

It should also be noted that ${\mu }_{03}$ and ${\mu }_{04}$ can be obtained on exchanging ${\theta }_1$ by ${\theta }_2$ in ${\mu }_{30}$ and ${\mu }_{40}$ and visa-versa.