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Minimax optimal sequential hypothesis tests for Markov processes. (English) Zbl 1460.62131
The paper deals with a generalization of the Kiefer and Weiss problem of constructing a sequential test that, in addition to meeting the targeted error probabilities under the hypotheses, minimizes the maximum expected run length over all feasible distributions [A. Dvoretzky et al., Ann. Math. Stat. 24, 254–264 (1953; Zbl 0050.14803); J. Kiefer and L. Weiss, Ann. Math. Stat. 28, 57–74 (1957; Zbl 0079.35406)].
A well-established drawback of sequential hypothesis tests is that their higher efficiency depends critically on the assumption that the process generating the observations indeed follows the assumed model. In the paper, under mild assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. 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. 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. Numerical examples for minimax optimal sequential tests under different uncertainties illustrate the theoretical results.
62L10 Sequential statistical analysis
62C20 Minimax procedures in statistical decision theory
62F35 Robustness and adaptive procedures (parametric inference)
62M02 Markov processes: hypothesis testing
Full Text: DOI Euclid
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