# Hypothesis testing with e-values

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 Papers

• Working Paper 1. "Non-algorithmic theory of randomness" by Vladimir Vovk (also published as an arXiv technical report).
This paper proposes an alternative language for expressing results of the algorithmic theory of randomness. The language is more precise in that it does not involve unspecified additive or multiplicative constants, making mathematical results, in principle, applicable in practice. In particular, it introduces e-values as a non-algorithmic counterpart of Levin-type deficiency of randomness. To appear 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. Springer, 2020.
• Working Paper 2. "Combining e-values and p-values" by Vladimir Vovk and Ruodu Wang (also published as an arXiv technical report).
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.
• Working Paper 3. "True and false 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. We prove an optimality result for our main procedure and demonstrate advantages of our methods over standard methods using simulated and real-world datasets.
• 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. Our simulation studies and theoretical considerations suggest that, under this assumption, our new algorithms are superior to the known algorithms using independent p-values and to our recent algorithms using e-values that are not necessarily independent.