Envelope Estimator of Multivariate Means

• Activity Number: 292
• Type: Contributed
• Date/Time: Tuesday, July 31, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Section on Risk Analysis
• Author(s): Zhihua Su*+ and Dennis Cook
• Companies: University of Minnesota and University of Minnesota
• Keywords: Envelope model ; James-Stein estimator ; dimension reduction ; Grassmann manifold ; reducing subspace

Abstract

Estimating the multivariate mean has been a long-standing problem. James and Stein (1961) proposed a shrinkage estimator that dominates the sample mean in quadratic loss. Since then, many works (e.g. Baranchik 1970, Berger 1976, Brown et al. 2011) emerged on searching estimators that are admissible, minimax, or with other desirable properties. We approach the problem from a entirely different perspective, the envelope point of view. Following the spirit of the envelope model introduced by Cook et al. (2010), we link the mean with the error structure. We assume that the mean is orthogonal to some eigenvectors of the error covariance matrix, which will happen with probability 1 when the dimension is large. Then if the eigenvectors are associated with large eigenvalues, the envelope estimator will achieve smaller risk. Numerical studies show that the risk of the envelope estimator can be much smaller than the Jame-Stein estimator.