On predictive density estimation for location families under integrated squared error loss

Abstract

Our investigation concerns the estimation of predictive densities and a study of efficiency as measured by the frequentist risk of such predictive densities with integrated squared error loss. Our findings relate to a d -variate spherically symmetric observable X ∼ p X ( ‖ x − μ ‖ 2 ) and the objective of estimating the density of Y ∼ q Y ( ‖ y − μ ‖ 2 ) based on X . We describe Bayes estimation, minimum risk equivariant estimation (MRE), and minimax estimation. We focus on the risk performance of the benchmark minimum risk equivariant estimator, plug-in estimators, and plug-in type estimators with expanded scale. For the multivariate normal case, we make use of a duality result with a point estimation problem bringing into play reflected normal loss. In three or more dimensions (i.e., d ≥ 3 ), we show that the MRE predictive density estimator is inadmissible and provide dominating estimators. This brings into play Stein-type results for estimating a multivariate normal mean with a loss which is a concave and increasing function of ‖ μ ˆ − μ ‖ 2 . We also study the phenomenon of improvement on the plug-in density estimator of the form q Y ( ‖ y − a X ‖ 2 ) , 0 < a ≤ 1 , by a subclass of scale expansions 1 c d q Y ( ‖ ( y − a X ) / c ‖ 2 ) with c > 1 , showing in some cases, inevitably for large enough d , that all choices c > 1 are dominating estimators. Extensions are obtained for scale mixture of normals including a general inadmissibility result of the MRE estimator for d ≥ 3 .