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Estimation of high dimensional predi...

Xu, Xinyi.

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Estimation of high dimensional predictive densities.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Estimation of high dimensional predictive densities./
作者:	Xu, Xinyi.
面頁冊數:	65 p.
附註:	Source: Dissertation Abstracts International, Volume: 66-06, Section: B, page: 3216.
Contained By:	Dissertation Abstracts International66-06B.
標題:	Statistics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3179839
ISBN:	0542201119

Estimation of high dimensional predictive densities.
Xu, Xinyi.

Estimation of high dimensional predictive densities. - 65 p.

Source: Dissertation Abstracts International, Volume: 66-06, Section: B, page: 3216.

Thesis (Ph.D.)--University of Pennsylvania, 2005.

Let X | mu &sim; Np(mu, vxI) and Y | mu &sim; Np(mu, vyI) 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 p(y | x) for Y that is close to p( y | mu) as measured by expected Kullback-Leibler loss. We connect this problem with estimating a multivariate normal mean under quadratic loss by establishing a new link identity, which explains striking similarities between these two problems and can be used to obtain analogous unifying theories of minimaxity and admissibility for density prediction. The natural straw man for this problem is the Bayesian predictive density pU( y | x) under the uniform prior pi U(mu) &equiv; 1, which is best invariant, minimax and admissible when p = 1 or 2. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal density that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate pU( y | x) including Bayes predictive densities under superharmonic priors. We also prove that all admissible procedures for this problem have to be formal Bayes rules, and that the conditions in Brown and Hwang (1982) are sufficient for a formal Bayes rule to be admissible. Furthermore, we show that many of these results can be extended to the linear regression setting with slight modifications. The shrinkage and multiple shrinkage properties of Bayesian predictive densities are also discussed.

ISBN: 0542201119Subjects--Topical Terms:

517247
Statistics.

Estimation of high dimensional predictive densities.
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520 $a Let X | mu &sim; Np(mu, vxI) and Y | mu &sim; Np(mu, vyI) 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 p(y | x) for Y that is close to p( y | mu) as measured by expected Kullback-Leibler loss. We connect this problem with estimating a multivariate normal mean under quadratic loss by establishing a new link identity, which explains striking similarities between these two problems and can be used to obtain analogous unifying theories of minimaxity and admissibility for density prediction. The natural straw man for this problem is the Bayesian predictive density pU( y | x) under the uniform prior pi U(mu) &equiv; 1, which is best invariant, minimax and admissible when p = 1 or 2. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal density that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate pU( y | x) including Bayes predictive densities under superharmonic priors. We also prove that all admissible procedures for this problem have to be formal Bayes rules, and that the conditions in Brown and Hwang (1982) are sufficient for a formal Bayes rule to be admissible. Furthermore, we show that many of these results can be extended to the linear regression setting with slight modifications. The shrinkage and multiple shrinkage properties of Bayesian predictive densities are also discussed.
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