Applications Of Bivariate Poisson Regression Model And Its Generalized Model

There have been a great development for Bivariate continuous data analysis by using Bivariate continuous distributions. however, because of the complicated forms of joint probability function, Bivariate discrete distributions, especial the Bivariate Poisson distribution which is the most important distribution of Bivariate discrete distributions, have few applications.In order to deal with Bivariate discrete data, there are two methods:first, we deal with discrete count data approximately by using continuous normal distribution. but, there often will be a great error; and the second one,on the independent assumption, we treat it as the double independent Poisson distribution data. If there exist dependence between in the count data, this method is also not appropriate.Because of the development of computer performance, there have been a gradual increase in the applications of Bivariate Poisson models and its extensions. Bivariate Poisson distribution is provided by Holgate (1964), its joint distribution can be gener-ated in many ways(see Kocherlakota & Kocherlakota(1992), here, we use the trivari-ate reduction method which have been provided in Holgate (1964),Kocherlakota &Kocherlakota(1992). If we let Wi,i=1,2,3 are independent random variables and Wi～Poisson(λi), i=1,2,3,Let N1=W1+W3 and N2=W2+W3, then the random variables N1,N2 follow jointly a Bivariate Poisson distribution: (N1,N2)～BP(λ1,λ2,λ3), We can give the joint probability functionIt have some useful properties:(1). From marginal means we known the marginal random variables are all to be Poisson with parametersλi+λ3,i=1,2.(2). Because soλ3 is a measure of the dependence for random count variables.(3). Let N=N1+N2, we can get the variance is and the mean is E(N)=E(N1+N2)=λ1+λ2+2λ3, so Var(N)> EN, and it is overdispersion.We can summarize the improvement of Bivariate Poisson distribution repalacing the independence assumption:(1). Bivariate Poisson distribution considered the correlation of count random variables;(2). Because most of the observed data is overdispersion, and from the properties of Bivariate Poisson distribution we know that Bivariate Poisson distribution is more suitable for data than double Poisson(it is equaldispersion);(3). Bivariate Poisson distribution can improve model fit and predictions of out-comes.Otherwise, this article also summarizes other properties, for instance, development of Bivariate Poisson, multivariate distributions, estimation of the parameters and so on.As the extension of Bivariate Poisson distribution, Zero-Inflated Bivariate Poisson distribution considered the large proportion of (0,0), it can give a useful describing for this data.Taken from Wang et al(2003),we obtain its joint probability function is where n1, n2 can't both equal to zero and 0< p< 1.From Kocherlakota & Kocherlakota(2001) we know, If we introduce the Bivari-ate Poisson and its extension to Model, we need to make regression for the param-eters. we follow the regressive method from Denuit(2007),like univariate Poisson regression, loglinear function of explanatory variables. we learn from Kocherlakota & Kocherlakota(2001),Morata(2009),Bivariate Poisson Regression Model is where i=1,2,3,..n,βj= (β1j,β2j,…βpj)′,j=1,2,3 are corresponding vectors of regression coefficients. and are vectors of explanatory variables.Learn from Walhin(2001),Wang et al(2003),Zero-Inflated Bivariate Regres-sion is(1). Take a logistic function of covariates Z with coefficients a as the parameter p and the Model is(2). we suppose that the logistic component concerning p is fixed and the Poisson parameterλi are related to the set of covariates, so the Model isthis article summarize the methods of estimated parameters of Regression Mod-els. About the estimations of the parameters of Regression Models, there have many methods. Numerical methods for maximizing the likelihood, Newton-Raphson itera-tive, have been used in many articles(see Jung & Winkelmann(1993),Kocherlakota & Kocherlakota(2001)); Ho & Singer(2001)used a generalized least squares method; EM algorithm to obtain maximum likelihood estimations are also used to estimate the parameters(see Karlis& Ntzoufras(2003)). otherwise, From a Bayesian point of view, MCMC method are also used to estimated parameters(see Karlis & Meligkotsidou(20 05)).In the last part, we summarized the applications of Regression Models in many fields,such as sports data analysis, ruin theory, labor mobility, especially in insurance.