Open Access
2024 Merging sequential e-values via martingales
Vladimir Vovk, Ruodu Wang
Author Affiliations +
Electron. J. Statist. 18(1): 1185-1205 (2024). DOI: 10.1214/24-EJS2228


We study the problem of merging sequential or independent e-values into one e-value or e-process. We describe a class of e-value merging functions via martingales and show that it dominates all merging methods for sequential e-values. All admissible methods for constructing e-processes can also be obtained in this way. In the case of merging independent e-values, the situation becomes much more complicated, and we provide a general class of such merging functions based on martingales applied to reordered data.

Funding Statement

The first author was supported by Amazon, Stena Line, and Mitie. The second author was supported in part by grants CRC-2022-00141 and RGPIN-2018-03823 from the Natural Sciences and Engineering Research Council of Canada.


We are grateful to the anonymous referees for helpful comments. For the role of Zhenyuan Zhang, see footnote 2.


Download Citation

Vladimir Vovk. Ruodu Wang. "Merging sequential e-values via martingales." Electron. J. Statist. 18 (1) 1185 - 1205, 2024.


Received: 1 August 2023; Published: 2024
First available in Project Euclid: 13 March 2024

arXiv: 2007.06382
Digital Object Identifier: 10.1214/24-EJS2228

Primary: 60G42 , 62G10
Secondary: 62C15

Keywords: Admissibility , Anytime validity , betting scores , e-processes , merging functions

Vol.18 • No. 1 • 2024
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