We meet in the literature the bivariate Poisson distribution put in evidence by Berkhout and Plug. From this distribution, Elion et al. put in evidence the bivariate weighted Poisson distribution like crossing of two univariate weighted Poisson distributions. The structure of the covariance of this bivariate weighted Poisson distribution has been put not again in evidence in the literature. Thus, in this paper, we remedy this hiatus. The overdispersion, underdispersion and the equidispersion will be valued with the help of the Fisher dispersion index for multivariate count distributions introduces by Kokonendji et al. An illustrative example based on the Aleurodicus data is presented.
Abstract. In the literature, there are two probabilistic models of bivariate Poisson : the model according to Holgate and the model according to Berkhout and Plug. These two models express themselves by their probability mass function. The model of Holgate puts in evidence a strictly positive correlation, which is not always realistic. To remedy this problem, Berkhout and Plug proposed a bivariate Poisson distribution accepting the correlation as well negative, equal to zero, that positive. In this paper, we show that these models are nearly everywhere asymptotically equal. From this survey that the φ-divergence converges toward zero, both models are therefore nearly everywhere equal. Also, the model of Holgate converges toward the one of Berkhout and Plug. Some graphs will be presented for illustrating this comparison.
In the recent statistical literature, the univariate Poisson distribution has been generalized by many authors, among them: the univariate weighted Poisson distribution [13], the generalized univariate Poisson distribution [7], the bivariate Poisson distribution according to Holgate [11], the bivariate Poisson distribution according to Lakshminarayana, Pandit and Srinivasa Rao [15], the bivariate Poisson distribution according to Berkhout and Plug [4], the bivariate weighted Poisson distribution according to Elion et al. [8] and the generalized bivariate Poisson distribution according to Famoye [9]. In this paper, We highlight the weighted bivariate Poisson distribution and show that it is the synthesis of all the bivariate Poisson distributions which, under certain conditions, converge in distribution towards the bivariate Poisson distribution according to Berkhout and Plug [4] which can be considered like the standard distribution in N2 as is the univariate Poisson distribution in N.
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