Optimized combination of independent or simultaneous e-values

TL;DR

This paper introduces an optimized method for combining e-values, maintaining validity even when tuning parameters are data-driven, applicable to independent and dependent e-variables, with an improved test based on symmetric polynomials.

Contribution

It presents a novel class of combined e-values that remain valid under data-driven tuning, extending to dependent structures, and proposes an enhanced combination test.

Findings

Validated the robustness of the optimized e-value combination method.

Extended the validity to a new class called simultaneous e-variables.

Developed an improved combination test using elementary symmetric polynomials.

Abstract

We show that a class of optimized e-value combinations, arising from a standard construction of e-processes, remains valid even when the tuning parameter is optimized based on the data. This result holds for independent e-values, and, more generally, for a new class called simultaneous e-variables, whose dependence structure lies between independence and sequential validity. We further propose an improved combination test for such e-values based on elementary symmetric polynomials.