Testing statistical hypothesis is usually done in sciences using p-values. In this project we promote using e-values, which are Bayes factors stripped of their Bayesian content. In some respects they are more convenient: e.g., the arithmetic mean of e-values is again an e-value, whereas merging p-values is more difficult. To a large degree, we are motivated by the algorithmic theory of randomness, which has both p-tests (introduced by Per Martin-Löf) and e-tests (introduced by Leonid Levin).

- Working Paper 1.
"Non-algorithmic theory of randomness"
by Vladimir Vovk
(also published as an arXiv technical report).
This paper briefly introduces e-values as a non-algorithmic counterpart of Levin-type deficiency of randomness. Published in:

*Fields of Logic and Computation III: Essays Dedicated to Yuri Gurevich on the Occasion of His 80th Birthday*, ed. by Andreas Blass, Patrick Cégilski, Nachum Dershowitz, Manfred Droste, and Bernd Finkbeiner.*Lecture Notes in Computer Science*, volume 12180, pages 323–340. Springer, 2020. - Working Paper 2.
"E-values: Calibration, combination, and applications"
by Vladimir Vovk and Ruodu Wang
(also published as an arXiv technical report).
We introduce e-values in their pure form to the statistical community. In particular, we demonstrate that e-values are often mathematically more tractable and develop procedures using e-values for multiple testing of a single hypothesis and testing multiple hypotheses. Published in the

*Annals of Statistics***49**:1736–1754 (2021). - Working Paper 3.
"Confidence and discoveries with e-values"
by Vladimir Vovk and Ruodu Wang
(also published as an arXiv technical report).
The topic of this paper is multiple hypothesis testing based on e-values. Using e-values instead of p-values leads to simple and efficient procedures that control the number of false discoveries under arbitrary dependence of the base e-values. Published in

*Statistical Science***38**:329–354 (2023). - Working Paper 4.
"True and false discoveries with independent e-values"
by Vladimir Vovk and Ruodu Wang
(also published as an arXiv technical report).
In this note we use e-values in the context of multiple hypothesis testing assuming that the base tests produce independent e-values.

- Working Paper 5.
"Merging sequential e-values via martingales"
by Vladimir Vovk and Ruodu Wang
(also published as an arXiv technical report).
We describe a general class of e-value merging functions via martingales and prove its optimality in a few senses. We also describe a general class of merging functions for independent e-values. To appear in the

*Electronic Journal of Statistics*. - Working Paper 6.
"Admissible ways of merging p-values under arbitrary dependence"
by Vladimir Vovk, Bin Wang, and Ruodu Wang
(also published as an arXiv technical report).
This paper studies the admissibility (in Wald's sense) of p-merging functions and their domination structure, without any assumptions on the dependence structure of the input p-values. As a technical tool we use the notion of e-values. Published in the

*Annals of Statistics***50**:351–375 (2022). - Working Paper 7.
"A note on data splitting with e-values"
by Vladimir Vovk
(also published as an arXiv technical report).
This note reanalyzes Cox's idealized example of testing with data splitting using e-values instead of Cox's p-values. Using e-values achieves inferential reproducibility, whose absence was the main drawback of Cox's method. This is an online appendix to my comment on Glenn Shafer's discussion paper "Testing by betting".

- Working Paper 8.
"Comment on Glenn Shafer's 'Testing by betting'"
by Vladimir Vovk
(also published as an arXiv technical report).
This note is my comment on Glenn Shafer's discussion paper "Testing by betting" complemented by two more online appendices. In Appendix A I briefly discuss p-values and e-values as different extensions of Cournot's principle and then list some of their advantages and disadvantages.

- Working Paper 9.
"Efficiency of nonparametric e-tests"
by Vladimir Vovk and Ruodu Wang
(also published as an arXiv technical report).
The notion of an e-value has been recently proposed as a possible alternative to critical regions and p-values in statistical hypothesis testing. In this paper we introduce a simple analogue for e-values of Pitmanâ€™s asymptotic relative efficiency and apply it to three popular nonparametric tests. To appear in the

*New England Journal of Statistics in Data Science*. - Working Paper 10.
"Multiple testing in game-theoretic probability: pictures and questions"
by Vladimir Vovk
(also published as an arXiv technical report).
The usual way of testing probability forecasts in game-theoretic probability is via construction of test martingales. The standard assumption is that all forecasts are output by the same forecaster. In this paper I discuss possible extensions of this picture to testing probability forecasts output by several forecasters. Its main goal is to report results of preliminary simulation studies and list some directions of further research (in support of my planned talk in Oberwolfach, 5–10 May 2024).