On the Breakdown Properties of Some Multivariate M-Functionals

For probability distributions on IR, a detailed study of the breakdown properties of some multivariate M-functionals related to Tyler’s (1987a) “distribution-free” M-functional of scatter is given. These include a symmetrized version of Tyler’s M-functional of scatter, and the multivariate t M-functionals of location and scatter. It is shown that for “smooth” distributions, the (contamination) breakdown point of Tyler’s M-functional of scatter and of its symmetrized version are 1/q and 1− √ 1− 1/q, respectively. For the multivariate t M-functional which arises from the maximum likelihood estimate for the parameters of an elliptical t distribution on ν ≥ 1 degrees of freedom the breakdown point at smooth distributions is 1/(q + ν). Breakdown points are also obtained for general distributions, including empirical distributions. Finally, the sources of breakdown are investigated. It turns out that breakdown can only be caused by contaminating distributions that are concentrated near low-dimensional subspaces.