Log-optimal anytime-valid E-values

Abstract

We consider the problem of measuring statistical evidence against a composite null hypothesis. We base our approach on the concept of an E-value, which measures evidence by the multiplication factor achieved by engaging in bets that are fair under the null. We adopt the log-optimality criterion for choosing among all possible E-values, which was considered earlier for a fixed sample size. We extend these ideas to sequential testing under optional stopping, by revisiting anytime-valid E-values. Our main contribution is the formulation of a sequential log-optimality criterion. We study its properties, and work out examples analytically and computationally.