Time-sensitive anytime-valid testing

TL;DR

This paper introduces a time-sensitive testing framework that prioritizes early rejection in anytime-valid tests, optimizing the timing of conclusions while controlling error rates.

Contribution

It develops a novel betting-based approach incorporating time preferences, leading to optimal control solutions and connections to classical statistical criteria.

Findings

Formulates a Bellman representation for time-sensitive testing.

Identifies the optimal e-process for finite-horizon Neyman--Pearson problems.

Introduces the exponential-decay-optimal (EDO) criterion as a finite-time analogue of the growth-rate-optimal approach.

Abstract

Anytime-valid tests allow evidence to be checked during data collection: one can either continue testing or stop and reject the null while still controlling type-I error. Yet, in many applications rejection is useful only if it comes soon enough. We introduce a time-sensitive testing-by-betting framework that favours early rejection by assigning rewards to rejection times and maximising their expected value under a given alternative. This encompasses hard deadlines and softer time preferences. The resulting optimal control problem admits a Bellman representation in terms only of time and evidence against the null, rather than the full history. For hard deadlines, the simple-vs-simple case reduces to a finite-horizon Neyman--Pearson problem and identify the corresponding optimal e-process. Furthermore, we show that exponentially decaying rewards admit a stationary approximation, yielding the exponential-decay-optimal (EDO) criterion: a finite-time-scale counterpart to the classical growth-rate-optimal (GRO) viewpoint in anytime-valid statistics, with the GRO criterion recovered in the large-time-scale limit.