Betting on Bets: Anytime-Valid Tests for Stochastic Dominance

TL;DR

This paper introduces a new family of sequential, anytime-valid tests for stochastic dominance, enabling real-time monitoring of distributional differences with strong power and validity guarantees.

Contribution

It develops nonparametric e-processes for sequential testing of stochastic dominance, including higher-order cases, with empirical validation against existing methods.

Findings

Sequential tests are competitive in power with non-sequential tests.

The tests maintain validity under continuous monitoring.

Method extends to higher-order stochastic dominance.

Abstract

How can we monitor, in real time, whether one uncertain prospect has any upside over another? To answer this question, we develop a novel family of sequential, anytime-valid tests for stochastic dominance (SD; also known as stochastic ordering), a classical and popular notion for comparing entire distribution functions. The problem is distinct from the popular problem of testing for dominance in means, which would not capture distributional differences beyond the first moment. We first derive powerful, nonparametric e-processes that quantify evidence against the null hypothesis that one prospect is dominated by another. For first-order SD, these e-processes are constructed as a mixture of asymptotically growth-rate optimal e-variables and yield a test of power one. The approach further generalizes to sequential testing for SD beyond the first order, including any higher-order SD. Empirically, we demonstrate that the resulting sequential tests are competitive with existing non-sequential SD tests in terms of power, while achieving validity under continuous monitoring that existing methods do not. Finally, we sketch the complementary and challenging problem of testing the non-SD null hypothesis, which asks whether a prospect has a definite upside, and describe the conditions under which we can derive a nontrivial anytime-valid test.