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

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∼pX(‖x−μ‖2) and the objective of estimating the density of Y∼qY(‖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 ‖μˆ−μ‖2. We also study the phenomenon of improvement on the plug-in density estimator of the form qY(‖y−aX‖2),01, 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.