  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
\sqrtn-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.
