The asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions

• 成果类型: Article
• 署名作者: Magyar, Andrew F.; Tyler, David E.
• 署名单位: Rutgers University System; Rutgers University New Brunswick
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asu020
• 发表日期: 2014
• 页码: 673688
• 关键词: Principal component analysis; multivariate location; breakdown properties; M-ESTIMATORS; EFFICIENCY; angles

摘要

The asymptotic efficiency of the spatial sign covariance matrix relative to affine equivariant estimators of scatter is studied. In particular, the spatial sign covariance matrix is shown to be asymptotically inadmissible, i.e., the asymptotic covariance matrix of the consistency-corrected spatial sign covariance matrix is uniformly larger than that of its affine equivariant counterpart, namely Tyler's scatter matrix. Although the spatial sign covariance matrix has often been recommended when one is interested in principal components analysis, its inefficiency is shown to be most severe in situations where principal components are of greatest interest. Simulation shows that the inefficiency of the spatial sign covariance matrix also holds for small sample sizes, and that the asymptotic relative efficiency is a good approximation to the finite-sample efficiency for relatively modest sample sizes.