Sample heterogeneity and M-estimation

Necessary and sufficient conditions are given for the consistency and rates of consistency of M-estimators (based on non-decreasing score functions) in heterogeneous location models. Some consequences are presented, among which an optimality property of the sample median, within the class of all M-estimators, in symmetric and unimodal models. The corresponding necessary and sufficient conditions for heteroscedastic location models are discussed. The results are used to illuminate some aspects of estimation in heterogeneous models; particular attention is paid to the so-called expansive models, where part of the probability "escapes to infinity". (C) 2001 Elsevier Science B.V. All rights reserved. MSG: primary: 62F35, 62F12; secondary: 62F10, 60F05.