Consistency of completely outlier-adjusted simultaneous redescending M-estimators of location and scale

Abstract

This paper gives conditions for the consistency of simultaneous redescending M-estimators for location and scale. The consistency postulates the uniqueness of the parameters μ and σ, which are defined analogously to the estimations by using the population distribution function instead of the empirical one. The uniqueness of these parameters is no matter of course, because redescending ψ- and χ-functions, which define the parameters, cannot be chosen in a way that the parameters can be considered as the result of a common minimizing problem where the sum of ρ-functions of standardized residuals is to be minimized. The parameters arise from two minimizing problems where the result of one problem is a parameter of the other one. This can give different solutions. Proceeding from a symmetrical unimodal distribution and the usual symmetry assumptions for ψ and χ leads, in most but not in all cases, to the uniqueness of the parameters. Under this and some other assumptions, we can also prove the consistency of the according M-estimators, although these estimators are usually not unique even when the parameters are. The present article also serves as a basis for a forthcoming paper, which is concerned with a completely outlier-adjusted confidence interval for μ. So we introduce a ñ where data points far away from the bulk of the data are not counted at all.