Projection based scatter depth functions and associated scatter estimators

In this article, we study a class of projection based scatter depth functions proposed by Zuo [Y. Zuo, Robust location and scatter estimators in multivariate analysis, The Frontiers in Statistics, Imperial College Press, 2005. Invited book chapter to honor Peter Bickel on his 65th Birthday]. In order to use the depth function effectively, some favorable properties are suggested for the common scatter depth functions. We show that the proposed scatter depth totally satisfies these desirable properties and its sample version possess strong and 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 consistency of the sample scatter estimators are established. The finite sample performance of the related scatter estimator shows that it has a very high breakdown point and good efficiency.