Some Comments on the Izawa Bivariate Gamma Distribution
Richard A. Johnson, Department of Statistics, University of Wisconsin, USA
With an application to microarray data in mind, we consider the bivariate gamma distribution due to Izawa (1965). However, inference procedures are still lacking, perhaps because the density function contains a modified Bessel function of the first kind of order k and is not particularly tractable. Looking ahead to the microarray application, we focus our attention on the model which has equal shape parameters. We verify the conditions for the MLE to be asymptotically normal and also determine which elements of the Fisher information matrix can be calculated in closed form and which require numerical integration. The numerical details are non-trivial. Simulation studies illuminate the properties of maximum likelihood estimators. We also establish an asymptotic test for independence in this non-regular case. In the context of microarray data, Izawa's bivariate gamma model produces two-dimensional patterns for gene expressions that are similar to those in many applications. We entertain a hierarchical model where latent mean expression levels are first generated from the Izawa bivariate gamma distribution. Then, conditional on the mean levels, independent and identically distributed gamma variables produce the gene expression observations. Our hierarchical model implies a mean-variance relationship observed in many earlier studies. For gene-specific inference, we take an empirical Bayes approach to estimate the mode of a posterior distribution which we obtain in closed form. We conclude with an application to a well-studied data set. This is joint work with Sang-Hoon Cho.