Inadmissibility of the best equivariant predictive density in the unknown variance case

TL;DR

This paper demonstrates that the best equivariant predictive density is inadmissible for Gaussian vectors with unknown mean and variance, and introduces Bayesian methods that outperform it across all dimensions.

Contribution

It proves the inadmissibility of the best equivariant predictive density in the unknown variance case and proposes Bayesian predictive densities that dominate it.

Findings

Best equivariant predictive density is inadmissible for all dimensions.

Bayesian predictive densities can outperform the equivariant one.

Results hold in a nonasymptotic framework regardless of dimension.

Abstract

In this work, we are concerned with the estimation of the predictive density of a Gaussian random vector where both the mean and the variance are unknown. In such a context, we prove the inadmissibility of the best equivariant predictive density under the Kullback-Leibler risk in a nonasymptotic framework. Our result stands whatever the dimension d of the vector is, even when d<=2, which can be somewhat surprising compared to the known variance setting. We also propose a class of priors leading to a Bayesian predictive density that dominates the best equivariant one. Throughout the article, we give several elements that we believe are useful for establishing the parallel between the prediction and the estimation problems, as it was done in the known variance framework.