Improved minimax predictive densities under Kullback--Leibler loss

Abstract

Let $X|\mu \sim N_p(\mu ,v_{x}I)$ and $Y|\mu \sim N_p(\mu ,v_{y}I)$ be independent p-dimensional multivariate normal vectors with common unknown mean $\mu$. Based on only observing $X=x$, we consider the problem of obtaining a predictive density $\hat{p}(y|x)$ for $Y$ that is close to $p(y|\mu )$ as measured by expected Kullback--Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density ${\hat{p}}_{\mathrm{U}}(y|x)$ under the uniform prior ${\pi }_{\mathrm{U}}(\mu )\equiv 1$, which is best invariant and minimax. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate ${\hat{p}}_{\mathrm{U}}(y|x)$, including Bayes predictive densities under superharmonic priors. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described.

Published at http://dx.doi.org/10.1214/009053606000000155 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Simultaneous prediction of independent Poisson observables
• Nonsubjective priors via predictive relative entropy regret
• Asymptotically minimax Bayes predictive densities

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