Improved minimax predictive densities under Kullback--Leibler loss

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dc.creatorGeorge, Edward I.
dc.creatorLiang, Feng
dc.creatorXu, Xinyi
dc.date2006-05-16
dc.date.accessioned2026-07-07T08:07:48Z
dc.date.available2026-07-07T08:07:48Z
dc.descriptionLet $X| μ\sim N_p(μ,v_xI)$ and $Y| μ\sim N_p(μ,v_yI)$ 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 $\hat{p}(y| x)$ 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 $\hat{p}_{\mathrm{U}}(y| x)$ under the uniform prior $π_{\mathrm{U}}(μ)\equiv 1$, 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.
dc.descriptionPublished 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)
dc.identifierhttps://arxiv.org/abs/math/0605432
dc.identifierhttp://arxiv.org/abs/math/0605432
dc.identifierAnnals of Statistics 2006, Vol. 34, No. 1, 78-91
dc.identifierdoi:10.1214/009053606000000155
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/131052
dc.subjectStatistics Theory
dc.subject62C20 (Primary) 62C10, 62F15 (Secondary)
dc.titleImproved minimax predictive densities under Kullback--Leibler loss
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