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Improved minimax predictive densities under Kullback-Leibler loss. (English) Zbl 1091.62003
Summary: Let \(X\,|\,\mu\sim N_p(\mu,v_xI)\) and \(Y\,|\,\mu\sim N_p(\mu,v_yI)\) 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 \(\widehat p(y\,|\,x)\) for \(Y\) that is close to \(p(y\,|\,\mu)\) as measured by the expected Kullback-Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density \(\widehat p_U(y\,|\,x)\) under the uniform prior \(\pi_U(\mu)\equiv 1\), 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 \(\widehat p_U(y\,|\,x)\), 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.

MSC:
62C20 Minimax procedures in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
62F15 Bayesian inference
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