On Predictive Density Estimation under $\alpha$-divergence Loss

TL;DR

This paper investigates the efficiency of predictive density estimators under $ ext{alpha}$-divergence loss for normal models, extending previous results by identifying conditions where variance expansion improves estimation, with implications for robustness and dominance.

Contribution

It generalizes earlier findings on predictive density improvement from Kullback-Leibler loss to a broader class of $ ext{alpha}$-divergence losses, covering various dimensions, variances, and parameter restrictions.

Findings

Variance expansion can improve predictive densities under $ ext{alpha}$-divergence loss.

Results unify across different dimensions, variances, and loss functions.

Theoretical insights include robustness and dominance properties.

Abstract

Based on $X \sim N_d(\theta, \sigma^2_X I_d)$, we study the efficiency of predictive densities under $\alpha-$divergence loss $L_{\alpha}$ for estimating the density of $Y \sim N_d(\theta, \sigma^2_Y I_d)$. We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension $d$, the variances $\sigma^2_X$ and $\sigma^2_Y$, the choice of loss $L_{\alpha}$; $\alpha \in (-1,1)$. The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with $\theta \in \Theta \subset \mathbb{R}^d$. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.