On predictive density estimation with additional information

TL;DR

This paper investigates predictive density estimators for multivariate normal distributions with additional information on the difference of means, proposing improvements over standard methods and establishing dominance results under various loss functions.

Contribution

It introduces novel dominance results for predictive densities incorporating additional mean difference information, including Bayesian improvements and connections to skew-normal distributions.

Findings

Improved predictive densities over benchmark methods.

Bayesian dominance results under reverse Kullback-Leibler and KL losses.

Connections to skew-normal distributions and new distribution forms.

Abstract

Based on independently distributed $X_1 \sim N_p(\theta_1, \sigma^2_1 I_p)$ and $X_2 \sim N_p(\theta_2, \sigma^2_2 I_p)$, we consider the efficiency of various predictive density estimators for $Y_1 \sim N_p(\theta_1, \sigma^2_Y I_p)$, with the additional information $\theta_1 - \theta_2 \in A$ and known $\sigma^2_1, \sigma^2_2, \sigma^2_Y$. We provide improvements on benchmark predictive densities such as plug-in, the maximum likelihood, and the minimum risk equivariant predictive densities. Dominance results are obtained for $\alpha-$divergence losses and include Bayesian improvements for reverse Kullback-Leibler loss, and Kullback-Leibler (KL) loss in the univariate case ($p=1$). An ensemble of techniques are exploited, including variance expansion (for KL loss), point estimation duality, and concave inequalities. Representations for Bayesian predictive densities, and in particular for $\hat{q}_{\pi_{U,A}}$ associated with a uniform prior for $\theta=(\theta_1, \theta_2)$ truncated to $\{\theta \in \mathbb{R}^{2p}: \theta_1 - \theta_2 \in A \}$, are established and are used for the Bayesian dominance findings. Finally and interestingly, these Bayesian predictive densities also relate to skew-normal distributions, as well as new forms of such distributions.