Elsevier

Journal of Multivariate Analysis

Volume 142, December 2015, Pages 57-74
Journal of Multivariate Analysis

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

https://doi.org/10.1016/j.jmva.2015.07.013Get rights and content
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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 XpX(xμ2) and the objective of estimating the density of YqY(yμ2) 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., d3), 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 μˆμ2. We also study the phenomenon of improvement on the plug-in density estimator of the form qY(yaX2),0<a1, by a subclass of scale expansions 1cdqY((yaX)/c2) 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 d3.

AMS 2010 subject classifications

62C20
62C86
62F10
62F15
62F30

Keywords

Concave loss
Convolutions
Dominance
Frequentist risk
Inadmissibility
Integrated squared error
Minimax
Minimum risk equivariant
Loss function
Multivariate normal
Predictive density
Restricted parameter space
Scale mixture of normals
Stein estimation

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