Robust M estimation in the two-sample problem

Let (Equation presented) be two independent samples from a location-scale family, and suppose that the interest lies in robust M estimation of the location difference θ₂ − θ₁. It will be assumed that the two scale parameters are equal but unknown. The usual robust M estimator for the location shift is the difference of the robust M estimators for the two location parameters. In this article we propose an alternative method that uses a single estimating equation to generate the robust estimator of the location shift. This estimating equation is derived from a reparametrization of the two-sample problem introduced by Neyman and Scott. When the same Ψ function is used, the new and the usual M estimators of the location shift have the same influence functions and asymptotic variance, and hence the two types of M estimators share the same class of optimal Ψ functions. However, all members of the new class of optimal robust M estimators for the location shift have sample-wise breakdown point .5 and thus are preferable over the usual M estimators in the asymmetric case. (In the symmetric case the usual M estimators also have sample-wise breakdown point .5.) The essential idea is to use the orthogonality of the Neyman-Scott parametrization, and then to use a maximum breakdown point (but possibly low-efficiency) estimator for the nuisance parameter. Some theoretical breakdown points and numerical illustrations are given.