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Minimax Robust Detection: Classic Results and Recent Advances

Fauß, M. ; Zoubir, A. M. ; Poor, H. V. (2021):
Minimax Robust Detection: Classic Results and Recent Advances.
In: IEEE Transactions on Signal Processing, 69, pp. 2252-2283. IEEE, ISSN 1053-587X, e-ISSN 1941-0476,
DOI: 10.1109/TSP.2021.3061298,
[Article]

Abstract

This paper provides an in-depth overview of results and concepts in minimax robust hypothesis testing for two and multiple hypotheses, where the latter refers to the case of more than two hypotheses. It starts with an introduction to the subject, highlighting its connection to other areas of robust statistics and giving a brief recount of the most prominent developments over the last decades. Subsequently, the minimax principle is introduced and its strengths and limitations are discussed. The technical part of the paper first focuses on the two-hypothesis case. After briefly reviewing the basics of statistical hypothesis testing, uncertainty sets are introduced as a generic way of modeling distributional uncertainty and model mismatch. The design of minimax detectors is then shown to reduce to the problem of determining a pair of least favorable distributions, and three criteria for their characterization are presented and discussed. In this context, the importance of f-divergences in robust detection is highlighted and explained. Explicit expressions are given for least favorable distributions under three types of uncertainty: epsilon-contamination, probability density bands, and f-divergence balls. All of them are illustrated with examples and it is demonstrated how the properties of least favorable distributions translate to properties of the test statistics of the corresponding minimax detectors. The second part of the paper deals with the problem of robustly testing multiple hypotheses, starting with a discussion of why it is fundamentally different from the binary problem and cannot be solved in the same manner. Sequential detection is then introduced as a technique that has recently been shown to enable strictly minimax optimal tests in the multi-hypothesis case. A numerical example illustrates the technical results. Finally, the usefulness of robust detectors in practice is showcased using the example of ground penetrating radar, which has applications in landmine detection, archaeology and integrity testing. The paper concludes with an outlook on robust detection beyond the minimax principle and a brief summary of the presented material.

Item Type: Article
Erschienen: 2021
Creators: Fauß, M. ; Zoubir, A. M. ; Poor, H. V.
Title: Minimax Robust Detection: Classic Results and Recent Advances
Language: English
Abstract:

This paper provides an in-depth overview of results and concepts in minimax robust hypothesis testing for two and multiple hypotheses, where the latter refers to the case of more than two hypotheses. It starts with an introduction to the subject, highlighting its connection to other areas of robust statistics and giving a brief recount of the most prominent developments over the last decades. Subsequently, the minimax principle is introduced and its strengths and limitations are discussed. The technical part of the paper first focuses on the two-hypothesis case. After briefly reviewing the basics of statistical hypothesis testing, uncertainty sets are introduced as a generic way of modeling distributional uncertainty and model mismatch. The design of minimax detectors is then shown to reduce to the problem of determining a pair of least favorable distributions, and three criteria for their characterization are presented and discussed. In this context, the importance of f-divergences in robust detection is highlighted and explained. Explicit expressions are given for least favorable distributions under three types of uncertainty: epsilon-contamination, probability density bands, and f-divergence balls. All of them are illustrated with examples and it is demonstrated how the properties of least favorable distributions translate to properties of the test statistics of the corresponding minimax detectors. The second part of the paper deals with the problem of robustly testing multiple hypotheses, starting with a discussion of why it is fundamentally different from the binary problem and cannot be solved in the same manner. Sequential detection is then introduced as a technique that has recently been shown to enable strictly minimax optimal tests in the multi-hypothesis case. A numerical example illustrates the technical results. Finally, the usefulness of robust detectors in practice is showcased using the example of ground penetrating radar, which has applications in landmine detection, archaeology and integrity testing. The paper concludes with an outlook on robust detection beyond the minimax principle and a brief summary of the presented material.

Journal or Publication Title: IEEE Transactions on Signal Processing
Journal volume: 69
Publisher: IEEE
Divisions: 18 Department of Electrical Engineering and Information Technology
18 Department of Electrical Engineering and Information Technology > Institute for Telecommunications
18 Department of Electrical Engineering and Information Technology > Institute for Telecommunications > Signal Processing
Date Deposited: 01 Mar 2021 07:34
DOI: 10.1109/TSP.2021.3061298
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