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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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