Bayesian Inference for Lancaster Probabilities

Inference for bivariate distributions with fixed marginals is very important in applications. When a bayesian approach is followed, the problem of defining a (prior) distribution on a class of probabilities having given marginals arises. We consider the class of Lancaster distributions. It is a convex and compact set, so that any element may be represented as a mixture of extreme points. Therefore a prior distribution can be assigned to the Lancaster class by assuming the mixing measure as a random probability. We analyse in detail the Lancaster class with Gamma marginals. Choosing as mixing measure a Dirichlet process, the model turns out to be a Dirichlet process mixture model. Many quantities relevant for statistical purposes are linear functionals of the Dirichlet process. Posterior laws are determined; in order to approximate these laws a MCMC algorithm is suggested. Results of an example with simulated data are discussed.