On Predictive Density Estimation for Location Families under Integrated $L_2$ and $L_1$ Losses

TL;DR

This paper investigates predictive density estimation under integrated $L_2$ and $L_1$ losses for multivariate spherically symmetric distributions, revealing inadmissibility of certain estimators and proposing improved methods.

Contribution

It provides new inadmissibility results and dominance strategies for predictive density estimators in multivariate settings under $L_2$ and $L_1$ losses.

Findings

MRE estimator is inadmissible for $p \\geq 3$ under $L_2$ loss.

Scale expansion improves plug-in estimators in high dimensions.

Inadmissibility results extend to scale mixtures of normals and $L_1$ 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 $L_2$ and $L_1$ losses. Our findings relate to a $p-$variate spherically symmetric observable $X \sim p_X(\|x-\mu\|^2)$ and the objective of estimating the density of $Y \sim q_Y(\|y-\mu\|^2)$ based on $X$. For $L_2$ loss, 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 of more dimensions (i.e., $p \geq 3$), we show that the MRE estimator is inadmissible under $L_2$ loss 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 $\|\hat{\mu}-\mu\|^2$. We also study the phenomenon of improvement on the plug-in density estimator of the form $q_Y(\|y-aX\|^2)\,, 0<a \leq 1\,,$ by a subclass of scale expansions $\frac{1}{c^p} \, q_Y(\|(y -aX)/c \|^2)$ with $c>1$, showing in some cases, inevitably for large enough $p$, 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 $p \geq 3$. Finally, we describe and expand on analogous plug-in dominance results for spherically symmetric distributions with $p \geq 4$ under $L_1$ loss.