Improved minimax predictive densities under Kullback--Leibler loss TeX Export

@misc{citeulike:1595844, abstract = {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 \$\hat{p}(y| x)\$ for \$Y\$ that is close to \$p(y| \mu)\$ 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 \$\pi\_{\mathrm{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 \$\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.}, archivePrefix = {arXiv}, author = {George, Edward I. and Liang, Feng and Xu, Xinyi}, citeulike-article-id = {1595844}, citeulike-linkout-0 = {http://arxiv.org/abs/math/0605432}, citeulike-linkout-1 = {http://arxiv.org/pdf/math/0605432}, day = {16}, eprint = {math/0605432}, keywords = {density-estimation, multivariate-normal}, month = {May}, posted-at = {2007-08-27 10:01:19}, priority = {3}, title = {Improved minimax predictive densities under Kullback\&\#45;\&\#45;Leibler loss}, url = {http://arxiv.org/abs/math/0605432}, year = {2006} }

TY - ELEC ID - citeulike:1595844 L3 - citeulike-article-id:1595844 N2 - 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 $\hat{p}(y| x)$ for $Y$ that is close to $p(y| \mu)$ 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 $\pi_{\mathrm{U}}(\mu)\equiv 1$, 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 $\hat{p}_{\mathrm{U}}(y| x)$, 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. TI - Improved minimax predictive densities under Kullback--Leibler loss KW - density-estimation KW - multivariate-normal AU - George, Edward AU - Liang, Feng AU - Xu, Xinyi PY - 2006/May/16/ UR - http://arxiv.org/abs/math/0605432 ER -