scholarly journals A Note on the (Weighted) Bivariate Poisson Distribution

2021 ◽  
Vol 14 (1) ◽  
pp. 192-203
Author(s):  
R. Bidounga ◽  
P. C. Batsindila Nganga ◽  
L. Niéré ◽  
D. Mizère
Keyword(s):  
Poisson Distribution ◽  
Poisson Distributions ◽  
Standard Distribution ◽  
Bivariate Poisson Distribution

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.

On a characterization of Poisson distributions

1972 ◽  
Vol 9 (4) ◽  
pp. 852-856 ◽  
Author(s):  
J. Aczél
Keyword(s):  
Poisson Distribution ◽  
Random Variables ◽  
Mutual Dependence ◽  
Binomial Distributions ◽  
Destructive Process ◽  
Poisson Distributions ◽  
Bivariate Poisson Distribution

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

On a characterization of Poisson distributions

1972 ◽  
Vol 9 (04) ◽  
pp. 852-856 ◽  
Author(s):  
J. Aczél
Keyword(s):  
Poisson Distribution ◽  
Random Variables ◽  
Mutual Dependence ◽  
Binomial Distributions ◽  
Destructive Process ◽  
Poisson Distributions ◽  
Bivariate Poisson Distribution

The conjecture pronounced at the end of the paper of Srivastava and Srivastava (1970) is proved in this paper. It gives the following characterization of (bivariate) Poisson distributions. Suppose that items of two types have been observed certain numbers of times, but these original observations have been reduced due to a destructive process which is the product of two binomial distributions and that the probabilities of these reduced numbers are the same whether damaged or undamaged. Then the original random variables had a bivariate Poisson distribution with zero mutual dependence coefficient.

A Note on the Bivariate Poisson Distribution

1969 ◽  
Vol 23 (4) ◽  
pp. 32 ◽  
Author(s):  
M. A. Hamdan ◽  
H. A. Al-Bayyati
Keyword(s):  
Poisson Distribution ◽  
Bivariate Poisson Distribution

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

2020 ◽  
Vol 9 (6) ◽  
pp. 1
Author(s):  
Rufin Bidounga ◽  
Evrand Giles Brunel Mandangui Maloumbi ◽  
Réolie Foxie Mizélé Kitoti ◽  
Dominique Mizère
Keyword(s):  
Probability Distribution ◽  
Poisson Distribution ◽  
Generalized Linear Models ◽  
Generating Functions ◽  
Linear Models ◽  
Double Series ◽  
Bivariate Distribution ◽  
Probability Generating Functions ◽  
Bivariate Poisson Distribution ◽  
Bivariate Function

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 overcome 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 effect 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.

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