The New Bivariate Conway-Maxwell-Poisson Distribution Obtained by the Crossing Method

Kimberly et al. had proposed in 2016 a bivariate function as a bivariate Conway-Maxwell-Poisson distribution (COM-Poisson) using the generalized bivariate Poisson distribution and the probability generating functions of the follow distributions: bivariate bernoulli, bivariate Poisson, bivariate geometric and bivariate binomial. By examining this paper we have shown that this bivariate function is constant and it double series is divergent, when it should have been 1. To over-come this deadlock, we propose a new bivariate Conway-Maxwell-Poisson distribution which is deﬁnetely a probability distribution based on the crossing method, method highlighted by Elion et al. in 2016 and revisited by Batsindila et al. and Mandangui et al. in 2019. And this is the purpose of this paper. To this bivariate distribution is attached two generalized linear models (GLM) whose resolution allows us to highlight, not only the independence between the variables forming the couple, but also the e ﬀ ect of the factors (or predictors) on these variables. The resulting correlation is negative, zero or positive depending on the values of a parameter; in particular for the bivariate Poisson distribution according to Berkhout and Plug. A simulation of data will be given at the end of the article to illustrate the model.

Indeed, X follow the weighted Poisson distribution variable, its characteristics are equal to (Kokonendji et al.,2008)

Approximation of Z(λ, ν)
We have the following approximations (Shmueli et al,2005) 1-When there is an integer N such as for n > N, λ n ν < 1, then Z(λ, ν) In the section (2), we will present the bivariate COM-Poisson distribution according to Kimberly et al.(2016) and in the section (3), the new bivariate COM-Poisson distribution. Finally, in the section (4), a simulation of the data will be carried out to illustrate the model. Kimberly et al. (2016) In this section, we will present only a few salient points of this study and for details see the article by Kimberly et al.(2016). Let (0, 0), (0, 1), (1, 0) and (1, 1) the values of a bivariate Bernoulli variable of probabilities p 00 , p 01 , p 10 and p 11 with p 00 + p 01 + p 10 + p 11 = 1.

The Bivariate COM-Poisson Distribution According to
Let following the multinomial expressions (Johnson et al., 1997;Kimberly et al.,2016) We have the following result
And the right part denoted D is equal to .
Given that G = D, we have the following result By posing the final result is given by n a=n−x−y n a, n − a − y, n − a − x, x + y + a − n p a 00 p n−a−y 10 p n−a−x 01 p x+y−n+a 11 ≡ 1.

Corollary
The following mass function proposed by Kimberly et al.(2016) as a bivariate COM-Poisson distribution is not a probability.
Indeed, we have Pr(Y = y, X = x) is divergent. This bivariate function is therefore not a probability because this double series should have been 1.

Definition: The Crossing Method
Let us consider two random variables X and Y which follow univariate COM-Poisson distribution of mass functions and The couple (X, Y) follows the bivariate COM-Poisson distribution if and only if it mass function is equal to (Elion et al., 2016;Batsindila et al., 2019 andMandangui et al.,2019) under the conditions and X is the response variable of the model (7) and Y that of the model (8), with t = (t 1 , t 2 , ..., t p ) the vector of predictors variables or factors. The expression (8) leads to the fact that P(Y = y/λ 2 , ν 2 ) = P(Y = y/X = x) is a conditional probability.
When η = 0, the variables X and Y are independent.

Characteristics
A simple application of proposition 1 from the work of Batsindila et al.(2019), gives us the following results Var(Y) = e 2λ 1 (e η −1) Z(λ 1 , ν 1 ) Z(λ 1 e η , ν 1 ) − 1 (10) The expressions (8) and (11) confirm that the variables X and Y are independent if and only if the parameter η = 0. The resulting correlation is negative, zero or positive depending on whether the value of the parameter η is negative, zero or positive in particular for the bivariate Poisson distribution according to Berkhout and Plug(2004).
Remarque 1. The variable X follow the COM-Poisson distribution, its characteristics have been given in the subsection (1.3) International Journal of Statistics and Probability Vol. 9, No. 6; 2020 3.3 Ratio of Successives Probabilities and Estimation of Parameters λ 1 , ν 1 , λ 2 , ν 2 Let The ratio of successives probabilities of the COM-Poisson distribution is equal to (Shmueli et al.,2005) we have by symmetry, ln p y−1 p y = −lnλ 2 + ν 2 lny.
When the plot of the set data lnx, ln p x−1 p x /x > 1 is adjusted by a straight line, then the random variable X follow a COM-Poisson distribution (Shmueli et al.,2005). Thanks to the distribution of large numbers, successive probabilities p x (p y ) can be replaced by successives frequencies f x ( f y ) associated with x(y). In this case, the expression ln f x−1 f x which is called log-ratio successives frequencies, will be used as a replacement for ln p x−1 p x , the response variable with lnx the explanatory variable in the model (12). Although the basic assumtions of homoscedasticity and independence of theses log-ratio are not established(Cf. Shmueli et al.,2005), the regression lines of (12) and (13) allow to calculate the estimators of λ 1 , ν 1 , λ 2 , ν 2 noted respectively λ 1 , ν 1 , λ 2 , ν 2 .

The Log Likelihood Estimation of Parameter β 1 , β 2 et η
Let consider the distributions P(X = x/λ 1 , ν 1 ), P(Y = y/λ 2 , ν 2 ) and generalized linear models (7) and (8), we will calculate the maximum likelihood estimators of the parameters β 1 , β 2 and η in order to be able to highlight, not only the independence between the variables X and Y but also the effect of factors on these variables.

The Log Likelihood Function
The log likelihood function of the bivariate COM-Poisson distribution is equal to By the replacement in the expressions (7) and (8) we have: after development we find 3.4.2 Parameters Estimate β 1 , β 2 et η The parameters β 1 , β 2 and η will be estimated by the maximum likelihood method. Either a sample (x i , y i ) of size n values of the couple of variables (X, Y) of probability density P.
The likelihood of observing this sample is equal to P(x i , y i , λ 1 , ν 1 , λ 2 , ν 2 ) International Journal of Statistics and Probability Vol. 9, No. 6;2020 By applying the logarithm, we have: lnP(x i , y i , λ 1 , ν 1 , λ 2 , ν 2 ), where the function P is defined by the expression (6). By using the expressions (7) and (8) We will use the function maxLik of the statistical computer software R to determine these estimators.

Data and Software Used
To illustrate this method, we will simulate the COM-Poisson data using the Statistics computer software R data processing software. Let us recall that the COM-Poisson distribution with two parameters, one canonical and the other dispersion.
The COM-Poisson data of the random variables X and Y simulated of size N = 50 are given in the table (1) and (2) Table 1. Variable X for λ 1 = 2 and ν 1 = 2 X 0 1 2 3 λ 1 ν 1 N Obs. 7 32 8 3 2 2 50 Table 2. Variable Y for λ 2 = 3 and ν 2 = 2 Y 0 1 2 3 4 λ 2 ν 2 N Obs. 8 20 16 5 1 3 2 50 Table 3. Simulated data of the Poisson variable t of parameter λ = 2 t 0 1 2 3 4 5 7 N Obs. 10 11 14 11 2 1 1 50  Hence the estimated parameters of the models (12) and (13) are (table (7))  (8), that at the level α = 5% of significance, p-value equal to 0.16153 is higher than α; therefore the coefficient η of estimateη = 0.27046 is null significantly; what confirm the independance between the variable X and Y. It also evident from this table that to the same level of significance, the coefficientsβ j ( j = 1, 2) was not null significantly because the p-value was smaller than α, what brings us to say that the factor t has the effect, and on the variable X and on the variable Y.

Conclusion
The bivariate function proposed by Kimberly et al.(2016) is not a probability distribution, so it cannot be used as a model to describe data. The bivariate COM-Poisson distribution that we have proposed in this paper is definetely a probability distribution. This distribution allowed us to highlight, not only the independence between the variables X and Y, but also the effect of the factors (or predictors) on these variables. The resulting covariance is negative, zero or positive depending on the values of a parameter; in particular for the bivariate Poisson distribution according to Berkhout and Plug(2004).