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Sequential multiple testing with generalized error control: an asymptotic optimality theory

25 August 2016
Yanglei Song
Georgios Fellouris
ArXiv (abs)PDFHTML
Abstract

The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least kkk mistakes, of any kind, is controlled. Under the second, the probabilities of at least k1k_1k1​ false positives and at least k2k_2k2​ false negatives are simultaneously controlled. For each formulation, the optimal expected sample size is characterized, to a first-order asymptotic approximation as the error probabilities go to 0, and a novel multiple testing procedure is proposed and shown to be 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 special case of i.i.d. observations in each stream, the gains of the proposed sequential procedures over fixed-sample size schemes are quantified.

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