Combining e-values and p-values

47 Pages Posted: 1 Jan 2020 Last revised: 26 May 2020

See all articles by Vladimir Vovk

Vladimir Vovk

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: December 14, 2019


Multiple testing of a single hypothesis and testing multiple hypotheses are usually done in terms of p-values. In this paper we replace p-values with their natural competitor, e-values, which are closely related to betting, Bayes factors, and likelihood ratios. We demonstrate that e-values are often mathematically more tractable; in particular, in multiple testing of a single hypothesis, e-values can be merged simply by averaging them. This allows us to develop ecient procedures using e-values for testing multiple hypotheses.

Keywords: Hypothesis testing, multiple hypothesis testing, Bayes factor, test martingale, admissible decisions

JEL Classification: C12

Suggested Citation

Vovk, Vladimir and Wang, Ruodu, Combining e-values and p-values (December 14, 2019). Available at SSRN: or

Ruodu Wang (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1

No contact information is available for Vladimir Vovk

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