A class of multivariate discrete distributions based on an approximate density in {GLMM}

Abstract

It is well known that the generalized linear mixed model is useful for analyzing the overdispersion and correlation structure for multivariate discrete data. In this paper, we derive an approximation of the density function for the generalized linear mixed model. This approximation is found to satisfy the properties of probability density function under some conditions. Therefore, this approximation can be regarded as a class of multivariate distributions. Estimation of the parameters in this class can be carried out by the maximum likelihood method. We give the likelihood ratio criteria for testing several covariance structures. Several simulation studies were also conducted for the Poisson log-normal model when the proposed density function is regarded as an approximate likelihood of the generalized linear mixed model.