Improved minimax predictive densities under Kullback–Leibler loss

The Annals of Statistics · 2006 · Volume 34 · Issue 1

Abstract

Let X|μ∼Np(μ,vxI) and Y|μ∼Np(μ,vyI) be independent p-dimensional multivariate normal vectors with common unknown mean μ. Based on only observing X=x, we consider the problem of obtaining a predictive density p̂(y|x) for Y that is close to p(y|μ) as measured by expected Kullback–Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density p̂U(y|x) under the uniform prior πU(μ)≡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 p̂U(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.

Keywords

62C10, 62C20, 62F15, Bayes rules, heat equation, inadmissibility, multiple shrinkage, multivariate normal, Prior distributions, shrinkage estimation, superharmonic marginals, superharmonic priors, unbiased estimate of risk

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