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.
Highlights
IntroductionAs a new field of mathematical statistics, originates from the pioneering works of John Tukey (1960) [1], Peter Huber (1964) [2], and Frank Hampel (1968) [3]
Efficient Robust and StableRobust statistics, as a new field of mathematical statistics, originates from the pioneering works of John Tukey (1960) [1], Peter Huber (1964) [2], and Frank Hampel (1968) [3].The term “robust” (Latin: strong, vigorous, sturdy, tough, powerful) was introduced into statistics by George Box (1953) [4].The reasons of research in this field of statistics are of a general mathematical nature: the conceptions of “optimality” and “stability” are mutually complementary in performance evaluation for almost all mathematical procedures, and the trade-off between them is often a sought goal.It is not rare that the performance of optimal solutions is rather sensitive to small violations of the assumed conditions of optimality
Since the term “stability” is overloaded in mathematics, the term “robustness” being its synonym is at present conventionally used in statistics and in optimal control theory: in general, it means the stability of statistical inference under uncontrolled violations of accepted distribution models
Summary
As a new field of mathematical statistics, originates from the pioneering works of John Tukey (1960) [1], Peter Huber (1964) [2], and Frank Hampel (1968) [3]. The least informative (favorable) distribution minimizing Fisher information over a certain distribution class is obtained with the subsequent use of the asymptotically optimal maximum likelihood parameter estimate for this distribution. Hampel’s local approach [6], we mostly emphasize its recently developed stable estimation branch with the originally posed variational calculus problems and rather prospective results on their application to robust statistics [8].
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