### Abstract

Statistical hypothesis tests are referred to as robust if they are insensitive to small, random deviations from the underlying model. For two hypotheses and fixed sample sizes, the robust testing is well studied and understood. However, few results exist for the case in which the number of samples is variable (i.e., sequential testing) and the number of hypotheses is larger than two (i.e., multiple hypothesis testing). This paper outlines a theory of minimax optimal sequential tests for multiple hypotheses under general distributional uncertainty. It is shown that, in analogy to the fixed sample size case, the minimax solution is an optimal test for the least favorable distributions, i.e., a test that optimally separates the most similar feasible distributions. The joint similarity of multiple distributions is shown to be determined by a weighted f-dissimilarity, whose corresponding function is given by the unique solution of a nonlinear integral equation and whose weights are given by the likelihood ratios of the past samples. As a consequence, the least favorable distributions depend on the past observations and the underlying random process becomes a Markov-process whose state variable coincides with the test statistic.

Original language | English (US) |
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Title of host publication | 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 1044-1046 |

Number of pages | 3 |

ISBN (Electronic) | 9781538665961 |

DOIs | |

State | Published - Feb 5 2019 |

Event | 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 - Monticello, United States Duration: Oct 2 2018 → Oct 5 2018 |

### Publication series

Name | 2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |
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### Conference

Conference | 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018 |
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Country | United States |

City | Monticello |

Period | 10/2/18 → 10/5/18 |

### All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Hardware and Architecture
- Signal Processing
- Energy Engineering and Power Technology
- Control and Optimization

### Keywords

- minimax procedures
- multiple hypothesis testing
- ro-bust hypothesis testing
- sequential analysis

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## Cite this

*2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018*(pp. 1044-1046). [8635956] (2018 56th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2018). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ALLERTON.2018.8635956