Theory & Methods: A Robust and Almost Fully Efficient M‐Estimator

Abstract

Simultaneous robust estimates of location and scale parameters are derived from a class of M‐estimating equations. A coefficient p (p > 0), which plays a role similar to that of a tuning constant in the theory of M‐estimation, determines the estimating equations. These estimating equations may be obtained as the gradient of a strictly convex criterion function. This article shows that the estimators are uniquely defined, asymptotically bi‐variate normal and have positive breakdown for some choices of p. When p = 0.12 and p = 0.3, the estimators are almost fully efficient for normal and exponential distributions: efficiencies with respect to the maximum likelihood estimators are 1.00 and 0.99, respectively. It is shown that the location estimator for known scale has the maximum breakdown point 0.5 independent of p, when the target model is symmetric. Also it is shown that the scale estimator has a positive breakdown point which depends on the choice of p. A simulation study finds that the proposed location estimator has smaller variance than the Hodges–Lehmann estimator, Huber’s minimax and bisquare M‐estimators.