Open Access
April 2024 The online closure principle
Lasse Fischer, Marta Bofill Roig, Werner Brannath
Author Affiliations +
Ann. Statist. 52(2): 817-841 (April 2024). DOI: 10.1214/24-AOS2370


The closure principle is fundamental in multiple testing and has been used to derive many efficient procedures with familywise error rate control. However, it is often unsuitable for modern research, which involves flexible multiple testing settings where not all hypotheses are known at the beginning of the evaluation. In this paper, we focus on online multiple testing where a possibly infinite sequence of hypotheses is tested over time. At each step, it must be decided on the current hypothesis without having any information about the hypotheses that have not been tested yet. Our main contribution is a general and stringent mathematical definition of online multiple testing and a new online closure principle, which ensures that the resulting closed procedure can be applied in the online setting. We prove that any familywise error rate controlling online procedure can be derived by this online closure principle and provide admissibility results. In addition, we demonstrate how shortcuts of these online closed procedures can be obtained under a suitable consonance property.

Funding Statement

L. Fischer acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project number 281474342/GRK2224/2.
M. Bofill Roig is a member of the EU Patient-centric clinical trial platform (EU-PEARL). EU-PEARL has received funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No. 853966. This Joint Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme and EFPIA and Children’s Tumor Foundation, Global Alliance for TB Drug Development nonprofit organization, Spring-works Therapeutics Inc. This publication reflects the authors’ views. Neither IMI nor the European Union, EFPIA or any associated partners are responsible for any use that may be made of the information contained herein.


The authors are grateful for the valuable comments of two anonymous referees and an Associate Editor, which have led to significant improvements of the paper. In addition, the authors would like to thank Jelle Goeman for a useful discussion.


Download Citation

Lasse Fischer. Marta Bofill Roig. Werner Brannath. "The online closure principle." Ann. Statist. 52 (2) 817 - 841, April 2024.


Received: 1 November 2022; Revised: 1 December 2023; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2370

Primary: 62L10

Keywords: closure principle , familywise error rate , online multiple testing

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.52 • No. 2 • April 2024
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