IMPROVED MULTIVARIATE NORMAL MEAN ESTIMATION WITH UNKNOWN COVARIANCE WHEN p IS GREATER THAN n

• 成果类型: Article
• 署名作者: Chetelat, Didier; Wells, Martin T.
• 署名单位: Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/12-AOS1067
• 发表日期: 2012
• 页码: 3137-3160
• 关键词: arbitrary quadratic loss; shrinkage estimation; wishart distribution; minimax estimators; singular wishart; matrix; distributions; vector; variables; inverses

摘要

We consider the problem of estimating the mean vector of a p-variate normal (theta, Sigma) distribution under invariant quadratic loss, (delta - theta)'Sigma(-1) (delta - theta), when the covariance is unknown. We propose a new class of estimators that dominate the usual estimator delta(0)(X) = X. The proposed estimators of theta depend upon X and an independent Wishart matrix S with n degrees of freedom, however, S is singular almost surely when p > n. The proof of domination involves the development of some new unbiased estimators of risk for the p > n setting. We also find some relationships between the amount of domination and the magnitudes of n and p.