Foundations and Trends® in Statistics > Vol 1 > Issue 1-2

Hypothesis Testing with E-values

By Aaditya Ramdas, Carnegie Mellon University, USA, aramdas@cmu.edu | Ruodu Wang, University of Waterloo, Canada, wang@uwaterloo.ca

 
Suggested Citation
Aaditya Ramdas and Ruodu Wang (2025), "Hypothesis Testing with E-values", Foundations and Trends® in Statistics: Vol. 1: No. 1-2, pp 1-390. http://dx.doi.org/10.1561/3600000002

Publication Date: 10 Sep 2025
© 2025 A. Ramdas and R. Wang
 
Subjects
Detection and estimation,  Information theory and statistics,  Risk analysis,  Game theoretic learning,  Nonparametric methods,  Online learning,  Probability and statistics,  Optimization: Game theory,  Hypothesis testing,  Principal-agent
 

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In this article:
Front Matter
1. Introduction
2. Validity: E-values under the null hypothesis
3. Efficiency: E-values under the alternative hypothesis
4. Post-hoc testing and decision making with e-values
5. The universal inference e-value
6. The log-optimal e-value and reverse information projection
7. Sequential anytime-valid inference using e-processes
8. Handling multiple e-values
9. False discovery rate control using compound e-values
10. Approximate and asymptotic e-values
11. Combining e-values and using e-values as weights
12. Using e-values to combine dependent p-values
13. False coverage rate control using e-confidence intervals
14. Further contrasts between e-values and p-values
15. Improved e-value thresholds under additional conditions
16. E-values and risk measures
Appendix
References

Abstract

This monograph is written to offer a humble, but unified, treatment of e-values in hypothesis testing. It is organized into three parts: Fundamental Concepts, Core Ideas, and Advanced Topics. The first part includes four chapters that introduce the basic concepts. The second part includes five chapters of core ideas such as universal inference, log-optimality, e-processes, operations on e-values, and e-values in multiple testing. The third part contains seven chapters of advanced topics. The monograph collates important results from a variety of modern papers on e-values and related concepts, and also contains many results not published elsewhere. It offers a coherent and comprehensive picture on a fast-growing research area, and is ready to use as the basis of a graduate course in statistics and related fields.

DOI:10.1561/3600000002

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ISBN: 978-1-63828-590-8
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Table of contents:
Front Matter
1. Introduction
2. Validity: E-values under the null hypothesis
3. Efficiency: E-values under the alternative hypothesis
4. Post-hoc testing and decision making with e-values
5. The universal inference e-value
6. The log-optimal e-value and reverse information projection
7. Sequential anytime-valid inference using e-processes
8. Handling multiple e-values
9. False discovery rate control using compound e-values
10. Approximate and asymptotic e-values
11. Combining e-values and using e-values as weights
12. Using e-values to combine dependent p-values
13. False coverage rate control using e-confidence intervals
14. Further contrasts between e-values and p-values
15. Improved e-value thresholds under additional conditions
16. E-values and risk measures
Appendix
References

Hypothesis Testing with E-values

An e-value is a nonnegative test statistic whose expected value is at most one under the null hypothesis. This monograph is written to offer a humble, but unified, treatment of e-values in hypothesis testing. The need for such a work at this time can be explained by at least four reasons: (a) e-values have been named, utilized, and studied as a stand-alone concept only in the last few years, and a large body of its potential users do not know what they are; (b) e-values are fundamental objects at the core of hypothesis testing and estimation, and they are both under-studied and under-utilized; (c) several application domains in the natural and social sciences would benefit from knowing and adopting methodologies based on e-values in certain contexts to improve statistical efficiency and scientific reproducibility; (d) there has been an explosion of exciting research over the past few years, and hence the time is ripe to collate resources in a self-contained and concise manner.

The monograph is organized into three parts: Fundamental Concepts, Core Ideas, and Advanced Topics. The first part includes four sections that introduce the basic concepts. The second part includes five sections of core ideas such as universal inference, log-optimality, e-processes, operations on e-values, and e-values in multiple testing. The third part contains seven sections of advanced topics. The monograph collates important results from a variety of modern papers on e-values and related concepts, and also contains many results not published elsewhere. It offers a coherent and comprehensive picture on a fast-growing research area, and is ready to use as the basis of a graduate course in statistics and related fields.

 
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