May 2024 Testing with p*-values: Between p-values, mid p-values, and e-values
Ruodu Wang
Author Affiliations +
Bernoulli 30(2): 1313-1346 (May 2024). DOI: 10.3150/23-BEJ1633


We introduce the notion of p*-values (p*-variables), which generalizes p-values (p-variables) in several senses. The new notion has four natural interpretations: operational, probabilistic, Bayesian, and frequentist. A main example of a p*-value is a mid p-value, which arises in the presence of discrete test statistics. A unified stochastic representation for p-values, mid p-values, and p*-values is obtained to illustrate the relationship between the three objects. We study several ways of merging arbitrarily dependent or independent p*-values into one p-value or p*-value. Admissible calibrators of p*-values to and from p-values and e-values are obtained with nice mathematical forms, revealing the role of p*-values as a bridge between p-values and e-values. The notion of p*-values becomes useful in many situations even if one is only interested in p-values, mid p-values, or e-values. In particular, deterministic tests based on p*-values can be applied to improve some classic methods for p-values and e-values.

Funding Statement

The author acknowledges financial support from and the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).


The author thanks an Editor, an Associate Editor, three anonymous referees, Ilmun Kim, Tiantian Mao, Marcel Nutz, Aaditya Ramdas, Vladimir Vovk, Qiuqi Wang, Qinyu Wu and Zhenyuan Zhang for helpful comments and technical discussions on an earlier version of the paper.


Download Citation

Ruodu Wang. "Testing with p*-values: Between p-values, mid p-values, and e-values." Bernoulli 30 (2) 1313 - 1346, May 2024.


Received: 1 October 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699554
Digital Object Identifier: 10.3150/23-BEJ1633

Keywords: arbitrary dependence , average of p-values , mid p-values , posterior predictive p-values , test martingale


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Vol.30 • No. 2 • May 2024
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