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Sequential multiple testing with generalized error control: an asymptotic optimality theory. (English) Zbl 1433.62244

Consider \(k\) multiple testing of hypotheses. Historically, in the early stages of its development the focus was on procedures that control the probability of at least one false positive (incorrectly rejected null). It is easily checked by examples that if the number of hypotheses \(k\) is large, this error control is too prohibitive, and less stringent error metrics have been proposed and used in many publications. For example, during the last two decades, procedures were developed to control such error metrics as (i) the expectation or quantiles of the false discovery proportion (proportion of false positives among the rejected nulls), and (ii) the generalized familywise error rate, that is, the probability of at least \(k\geq 1\) false positives. In many of the references dealing with such error metrics, fixed sample sizes have been used.
In this paper, the authors develop sequential procedures to control the following error metrics: (i) under the first one, the probability of at least \(k\) mistakes, of any kind, is controlled, and (ii) under the second one, the probabilities of at least \(k_1\) false positives and at least \(k_2\) false negatives are simultaneously controlled. Asymptotic (as the error probabilities go to 0) optimal procedures are found and the underlying optimal expected sample size is characterized to a first-order approximation, and the proposed multiple testing procedure is asymptotically efficient under every signal configuration. These results are established when the data streams for the various hypotheses are independent and each local log-likelihood ratio statistic satisfies a certain strong law of large numbers. In the i.i.d case for each stream, the gains of the proposed sequential procedure over the fixed-sample size schemes are quantified.

MSC:

62L10 Sequential statistical analysis
62J15 Paired and multiple comparisons; multiple testing
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