On improved predictive density estimation with parametric constraints

Abstract

We consider the problem of predictive density estimation for normal models under Kullback-Leibler loss (KL loss) when the parameter space is constrained to a convex set. More particularly, we assume that $X\sim {\cal N}_{p}(\mu,v_{x}I)$ is observed and that we wish to estimate the density of $Y\sim {\cal N}_{p}(\mu,v_{y}I)$ under KL loss when μ is restricted to the convex set C⊂ℝ^p. We show that the best unrestricted invariant predictive density estimator p̂_U is dominated by the Bayes estimator p̂_πC associated to the uniform prior π_C on C. We also study so called plug-in estimators, giving conditions under which domination of one estimator of the mean vector μ over another under the usual quadratic loss, translates into a domination result for certain corresponding plug-in density estimators under KL loss. Risk comparisons and domination results are also made for comparisons of plug-in estimators and Bayes predictive density estimators. Additionally, minimaxity and domination results are given for the cases where: (i) C is a cone, and (ii) C is a ball.