Open Access
February 2005 Depth weighted scatter estimators
Yijun Zuo, Hengjian Cui
Ann. Statist. 33(1): 381-413 (February 2005). DOI: 10.1214/009053604000000922


General depth weighted scatter estimators are introduced and investigated. For general depth functions, we find out that these affine equivariant scatter estimators are Fisher consistent and unbiased for a wide range of multivariate distributions, and show that the sample scatter estimators are strong and $\sqrt{n}$-consistent and asymptotically normal, and the influence functions of the estimators exist and are bounded in general. We then concentrate on a specific case of the general depth weighted scatter estimators, the projection depth weighted scatter estimators, which include as a special case the well-known Stahel–Donoho scatter estimator whose limiting distribution has long been open until this paper. Large sample behavior, including consistency and asymptotic normality, and efficiency and finite sample behavior, including breakdown point and relative efficiency of the sample projection depth weighted scatter estimators, are thoroughly investigated. The influence function and the maximum bias of the projection depth weighted scatter estimators are derived and examined. Unlike typical high-breakdown competitors, the projection depth weighted scatter estimators can integrate high breakdown point and high efficiency while enjoying a bounded-influence function and a moderate maximum bias curve. Comparisons with leading estimators on asymptotic relative efficiency and gross error sensitivity reveal that the projection depth weighted scatter estimators behave very well overall and, consequently, represent very favorable choices of affine equivariant multivariate scatter estimators.


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Yijun Zuo. Hengjian Cui. "Depth weighted scatter estimators." Ann. Statist. 33 (1) 381 - 413, February 2005.


Published: February 2005
First available in Project Euclid: 8 April 2005

zbMATH: 1065.62048
MathSciNet: MR2157807
Digital Object Identifier: 10.1214/009053604000000922

Primary: 62F35 , 62H12
Secondary: 62E20 , 62F12

Keywords: asymptotic normality , Breakdown point , consistency , depth , efficiency , influence function , maximum bias , projection depth , robustness , Scatter estimator

Rights: Copyright © 2005 Institute of Mathematical Statistics

Vol.33 • No. 1 • February 2005
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