Bayesian Predictive Density Estimation for a Chi-squared Model Using Information from a Normal Observation with Unknown Mean and Variance

In this paper, we consider the problem of estimating the density function of a Chi-squared variable on the basis of observations of another Chi-squared variable and a normal variable under the Kullback-Leibler divergence. We assume that these variables have a common unknown scale parameter and that the mean of the normal variable is also unknown. We compare the risk functions of two Bayesian predictive densities: one with respect to a hierarchical shrinkage prior and the other based on a noninformative prior. The hierarchical Bayesian predictive density depends on the normal variable while the Bayesian predictive density based on the noninformative prior does not. Sufficient conditions for the former to dominate the latter are obtained. These predictive densities are compared by simulation.