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.

中文翻译：

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Minimax稳健检测：经典结果和最新进展

本文概述了针对两个和多个假设的minimax稳健假设检验的结果和概念。首先介绍该主题，着重介绍该主题与稳健统计的其他领域的联系，并简要介绍最突出的发展。随后，介绍了极大极小原理，并讨论了其优点和局限性。本文的第一部分着重于两个假设的案例。在简要回顾了统计假设检验的基础之后，介绍了不确定性集作为建模分布不确定性的通用方法。然后显示了minimax检测器的设计，以减少确定一对最不利分布的问题，并讨论了其表征的不同标准。$ \ varepsilon $ -污染，概率密度带和 $ f $ -发散球。通过示例显示了这些最不利分布的特性如何转化为相应的minimax检测器的特性。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根本不同。然后引入顺序检测作为一种技术，该技术可以在多重假设的情况下设计严格最小极大值的最优检验。最后，以探地雷达为例，展示了坚固耐用的探测器在实践中的实用性。本文以超越最小极大值原理的鲁棒检测为展望，并对提出的材料进行了简要总结。