Highly Efficient Robust and Stable M -Estimates of Location

This article is partially a review and partially a contribution. The classical two approaches to robustness, Huber’s minimax and Hampel’s based on influence functions, are reviewed with the accent on distribution classes of a non-neighborhood nature. Mainly, attention is paid to the minimax Huber’s M -estimates of location designed for the classes with bounded quantiles and Meshalkin-Shurygin’s stable M -estimates. The contribution is focused on the comparative performance evaluation study of these estimates, together with the classical robust M -estimates under the normal, double-exponential (Laplace), Cauchy, and contaminated normal (Tukey gross error) distributions. The obtained results are as follows: (i) under the normal, double-exponential, Cauchy, and heavily-contaminated normal distributions, the proposed robust minimax M -estimates outperform the classical Huber’s and Hampel’s M -estimates in asymptotic efficiency; (ii) in the case of heavy-tailed double-exponential and Cauchy distributions, the Meshalkin-Shurygin’s radical stable M -estimate also outperforms the classical robust M -estimates; (iii) for moderately contaminated normal, the classical robust estimates slightly outperform the proposed minimax M -estimates. Several directions of future works are enlisted.