Projection based scatter depth functions and associated scatter estimators

A class of projection based scatter depth functions is introduced and studied. In order to use the depth function effectively, some favorable properties are suggested for the common scatter depth functions. We show the proposed scatter depth totally satisfies these desirable properties and its sample version possess strong and √ n uniform consistency. Under some regularity conditions, the limiting distribution of the empirical process of the scatter depth function is derived. We also found that the aforementioned depth functions assess the bounded influence functions. A maximum depth-based affine equivariant scatter estimator is induced. The limiting distributions as well as the strong and √ n consistency of the sample scatter estimators are establish. The finite sample performance of the related scatter estimator shows it has a very high breakdown point and appropriate efficiency. AMS 2000 Subject Classification: Primary 62G99 Secondary 60F15.