v vi statistics. Chapter 2 gives a new probability-free approach to constructing optimisation criteria in data analysis. Chapter 3 contains new results on robust minimax (in the Huber sense) estimation of location over the distribution classes with bounded variances and subranges, as well as for the classes of lattice distributions. Chapter 4 is confined to robust estimation of scale. Chapter 5 deals with robust regression and autoregression problems. Chapter 6 covers the particular case of L 1-norm estimation. Chapter 7 treats robust estimation of correlation. Chapter 8 introduces and discusses data analysis technologies, and Chapter 9 represents applications. We would like to express our deep appreciation to I. B. Chelpanov, E. P. Guilbo, B. T. Polyak, and Ya. Z. Tsypkin who attracted our attention to robust statistics. We are grateful to S. A. Aivazyan and L. D. Meshalkin for discussions and comments of our results at their seminars in the Central Institute of Economics and Mathematics of Russian Academy of Sciences (Moscow). We are very grateful to A. V. Kolchin for his great help in the preparation of this book. Finally, we highly appreciate V. Yu. Korolev and V. M. Zolotarev for their attention to our work and, in general, to such an important field of mathematical statistics as robustness.
In finite sample studies redescending M -estimators outperform bounded M -estimators (see for example, Andrews et al., 1972). Even though redescenders arise naturally out of the maximum likelihood approach if one uses very heavy-tailed models, the commonly used redescenders have been derived from purely heuristic considerations. Using a recent approach proposed by Shurygin, we studied the optimality of redescending M -estimators. We show that redescending M -estimator can be designed by applying a global minimax criterion to locally robust estimators, namely maximizing the minimum variance sensitivity of an estimator over a given class of densities. As a particular result, we proved that Smith's estimator, which is a compromise between Huber's skipped mean and Tukey's biweight, provides the guaranteed level of an estimator's variance sensitivity over the class of distribution densities with a bounded variance.
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