Improved minimax predictive densities under Kullback--Leibler loss

George, Edward I.; Liang, Feng; Xu, Xinyi

Improved minimax predictive densities under Kullback--Leibler loss

Authors

George, Edward I.

Liang, Feng

Xu, Xinyi

Description

Let $X| μ\sim N_p(μ,v_xI)$ 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.
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)