**Georgy Shevlyakov** †

Department of Applied Mathematics, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia; georgy.shevlyakov@phmf.spbstu.ru

† Prof. Georgy Shevlyakov passed away during the revision cycle of the manuscript, comments have been adressed by Dr. Maya Shevlyakova.

Received: 27 August 2020; Accepted: 30 December 2020; Published: 5 January 2021

**Abstract:** 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.

**Keywords:** robustness; minimax approach; stable estimation
