Huber’s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

Huber’s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with... A brief survey of former and recent results on Huber’s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber’s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavy-tailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber’s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber’s for heavy-tailed distributions and more efficient than Huber’s for short-tailed ones both in asymptotics and on finite samples. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Huber’s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

Huber’s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 2 (2005) 269–284 Huber’s Minimax Approach in Distribution Classes with Bounded Variances and Subranges with Applications to Robust Detection of Signals Georgy Shevlyakov, Kiseon Kim Department of Information and Communications, Gwangju Institute of Science and Technology (GIST), Oryong- dong, Buk-gu, Gwangju, 500-712, Republic of Korea (E-mail: shev@gist.ac.kr, kskim@gist.ac.kr) Abstract A brief survey of former and recent results on Huber’s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber’s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavy- tailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber’s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber’s for heavy-tailed distributions and more efficient than Huber’s for short-tailed ones both in asymptotics and on finite samples. Keywords Robustness, least informative distributions 2000 MR Subject Classification 62G35 1 Introduction 1.1 Preliminaries [13] About forty years has passed since publishing of the pioneer works of J.W. Tukey ,P.J. [4] [2] Huber , and F.R. Hampel . These outstanding personalities defined a new area of mathe- matical statistics called robust statistics which has been intensively developed since the sixties and is...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0235-x
Publisher site
See Article on Publisher Site

Abstract

A brief survey of former and recent results on Huber’s minimax approach in robust statistics is given. The least informative distributions minimizing Fisher information for location over several distribution classes with upper-bounded variances and subranges are written down. These least informative distributions are qualitatively different from classical Huber’s solution and have the following common structure: (i) with relatively small variances they are short-tailed, in particular normal; (ii) with relatively large variances they are heavy-tailed, in particular the Laplace; (iii) they are compromise with relatively moderate variances. These results allow to raise the efficiency of minimax robust procedures retaining high stability as compared to classical Huber’s procedure for contaminated normal populations. In application to signal detection problems, the proposed minimax detection rule has proved to be robust and close to Huber’s for heavy-tailed distributions and more efficient than Huber’s for short-tailed ones both in asymptotics and on finite samples.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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