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

The S-functionals of multivariate location and scatter, including the MVE-f unctionals, are known to be uniquely defined only at unimodal elliptically symmetric distributions. The goal of this paper is to establish the uniquen ess 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 th e 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 obtain ed for this class of multivariate functionals. Besides the S-functionals, t he class of multivariate M-functionals with auxiliary scale include the con strained M-functionals recently introduced by Kent and Tyler, as well as a new multivariate generalization of Yohai's MM-functionals.