  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.
