Information and communication technologies (ICT) have brought forth various useful tools and services, enabling another Internet-based industrial revolution over the last few decades

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.

This work considers algorithms of outlier detection based on the Chebyshev inequality. It compares these algorithms with such classical methods as Tukey’s boxplot, the N-sigma rule and its robust modifications based on MAD and FQ scale estimates. To adjust the parameters of the algorithms, a selection procedure is proposed based on the complete knowledge of the data distribution model. Areas of suboptimal parameters are also determined in case of incomplete knowledge of the distribution model. It is concluded that the direct use of the Chebyshev inequality implies the classical N-sigma rule. With the non-classical Chebyshev inequality, a robust outlier detection method is obtained, which slightly outperforms other considered algorithms.