## The Annals of Statistics

### Sequential multiple testing with generalized error control: An asymptotic optimality theory

#### Abstract

The sequential multiple testing problem is considered under two generalized error metrics. Under the first one, the probability of at least $k$ mistakes, of any kind, is controlled. Under the second, the probabilities of at least $k_{1}$ false positives and at least $k_{2}$ 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.

#### Article information

Source
Ann. Statist., Volume 47, Number 3 (2019), 1776-1803.

Dates
Revised: April 2018
First available in Project Euclid: 13 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.aos/1550026857

Digital Object Identifier
doi:10.1214/18-AOS1737

Mathematical Reviews number (MathSciNet)
MR3911130

Zentralblatt MATH identifier
07053526

Subjects
Primary: 62L10: Sequential analysis

#### Citation

Song, Yanglei; Fellouris, Georgios. Sequential multiple testing with generalized error control: An asymptotic optimality theory. Ann. Statist. 47 (2019), no. 3, 1776--1803. doi:10.1214/18-AOS1737. https://projecteuclid.org/euclid.aos/1550026857

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#### Supplemental materials

• Supplement to “Sequential multiple testing with generalized error control: An asymptotic optimality theory”. In the supplementary file, we present (i) more simulation studies, (ii) proofs of all results in this paper and (iii) additional technical lemmas.