Depth weighted scatter estimators

Abstract

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.