Regularity and Uniqueness for Constrained M-Estimates and Redescending M-Estimates

Abstract

Constrained M-estimates of multivariate location and scatter are found by finding the global minimum of an objective function subject to a constraint. They are related to redescending M-estimates of multivariate location and scatter since any stationary point of the objective function corresponds to such an M-estimate. Unfortunately, even for the population form of the estimator, that is, the constrained M-functional, the objective function may have multiple stationary points. In this paper, we give conditions under which the objective function is as well behaved as possible, in particular that it has at most one local minimum. To carry out this task, we introduce a class of distributions which we call "regular" distributions with respect to a particular objective function.