Betting on Bets: Anytime-Valid Tests for Stochastic Dominance

2026-04-23

AI Summary

This study addresses the challenge of determining, in real-time monitoring settings, whether one uncertain prospect stochastically dominates another in terms of its entire distribution—not merely their means. The work proposes the first sequential, anytime-valid testing procedure capable of assessing stochastic dominance (SD) of arbitrary order. Built upon nonparametric e-variables that are asymptotically growth-rate optimal under mixture constructions, the method constructs an e-process that dynamically accumulates evidence over continuous observations to reject the null hypothesis that one prospect is stochastically dominated by the other. The resulting test achieves high statistical power while maintaining strict time-uniform validity, matching the performance of state-of-the-art non-sequential tests and overcoming the longstanding limitation that traditional approaches cannot verify higher-order stochastic dominance in real time.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• stochastic dominance
• anytime-valid tests
• sequential testing
• distribution comparison
• nonparametric inference

Innovation

• anytime-valid test
• stochastic dominance
• e-process
• sequential testing
• nonparametric inference