Under mild Markov assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. First, the design of optimal sequential tests for simple hypotheses is revisited and it is shown that the partial derivatives of the corresponding cost function are closely related to the performance metrics of the underlying sequential test. Second, an implicit characterization of the least favorable distributions for a given testing policy is stated. By combining the results on optimal sequential tests and least favorable distributions, sufficient conditions for a sequential test to be minimax optimal under general distributional uncertainties are obtained. The cost function of the minimax optimal test is further identified as a generalized $f$-dissimilarity and the least favorable distributions as those that are most similar with respect to this dissimilarity. Numerical examples for minimax optimal sequential tests under different uncertainties illustrate the theoretical results.

中文翻译：

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马尔可夫过程的极小极大最优序列假设检验

在温和的马尔可夫假设下，推导出了分布不确定性下多个假设的严格最小最大最优性的充分条件。首先，重新审视了简单假设的最优顺序测试的设计，并表明相应成本函数的偏导数与基础顺序测试的性能指标密切相关。其次，陈述了给定测试策略的最不利分布的隐式特征。通过结合最优序列测试和最不利分布的结果，获得了在一般分布不确定性下使序列测试成为极小极大最优的充分条件。minimax 最优测试的成本函数被进一步确定为广义的 $f$-dissimilarity 和最不利的分布，即与这种差异最相似的分布。不同不确定度下的极小极大优化序列测试的数值例子说明了理论结果。