On the uniqueness of S-functionals and M-functionals under nonelliptical distributions

Abstract

The S-functionals of multivariate location and scatter, includingthe MVE-functionals, are known to be uniquely defined only at unimodal elliptically symmetric distributions. The goal of this paper is to establish the uniqueness of these functionals under broader classes of symmetric distributions. We also discuss some implications of the uniqueness of the functionals and give examples of striclty unimodal and symmetric distributions for which the MVE-functional is not uniquely defined. The uniqueness results for the S-functionals are obtained by embedding them within a more general class of functionals which we call the M-functionals with auxiliary scale. The uniqueness results of this paper are then obtained for this class of multivariate functionals. Besides the S-functionals, the class of multivariate M-functionals with auxiliary scale include the constrained M-functionals recently introduced by Kent and Tyler, as well as a new multivariate generalization of Yohai’s MM-functionals.