Minimax Robust Detection: Classic Results and Recent Advances
Minimax Robust Detection: Classic Results and Recent Advances
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DOI：
10.1109/tsp.2021.3061298
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发表时间：
2021-05
影响因子：
5.4
通讯作者：
Michael Fauss;A. Zoubir;H. Poor
中科院分区：
文献类型：
--
作者：
Michael Fauss;A. Zoubir;H. Poor
本文概述了两个和多个假设的极大极小稳健假设检验的结果和概念。它首先介绍了这一主题，强调了它与其他可靠统计领域的联系，并简要叙述了最突出的发展。接着，介绍了极大极小原理，并讨论了它的优点和局限性。论文的第一部分主要讨论双假设情况。在简要回顾了统计假设检验的基础知识之后，不确定性集作为一种建模分布不确定性的通用方法被引入。然后表明极大极小检测器的设计可简化为确定一对最不利分布的问题，并讨论了表征它们的不同准则。给出了三种不确定性类型下的最不利分布的显式表达式：$\varepsilon$-污染，概率密度带和$f$-发散球。通过实例，说明了这些最不利分布的性质如何转化为相应的极大极小检测器的性质。本文的第二部分讨论了稳健地检验多个假设的问题，首先讨论了为什么这与二元问题有本质的不同。然后介绍了顺序检测作为一种技术，使设计严格的极小极大最优测试在多假设的情况下。最后，以探地雷达为例，说明了鲁棒探测器的实用性。本文最后对超越极大极小原理的鲁棒检测进行了展望，并对所提出的材料进行了简要总结。
This paper provides an overview of results and concepts in minimax robust hypothesis testing for two and multiple 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. Subsequently, the minimax principle is introduced and its strengths and limitations are discussed. The first part of the paper 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. The design of minimax detectors is then shown to reduce to the problem of determining a pair of least favorable distributions, and different criteria for their characterization are discussed. Explicit expressions are given for least favorable distributions under three types of uncertainty: $\varepsilon$-contamination, probability density bands, and $f$-divergence balls. Using examples, it is shown how the properties of these least favorable distributions translate to properties 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 this is fundamentally different from the binary problem. Sequential detection is then introduced as a technique that enables the design of strictly minimax optimal tests in the multi-hypothesis case. Finally, the usefulness of robust detectors in practice is showcased using the example of ground penetrating radar. The paper concludes with an outlook on robust detection beyond the minimax principle and a brief summary of the presented material.