Huber-robust likelihood ratio tests for composite nulls and alternatives

2024-08-26

AI Summary

We address sequential testing of composite null and alternative hypotheses under ε-contaminated data, where an arbitrary fraction ε of observations may be adversarially corrupted. We propose the first e-value-based robust sequential testing framework. Methodologically, we (1) extend Huber’s robust estimation paradigm to composite sequential hypothesis testing; (2) introduce Robust Inverse Projection (RIPr), a novel technique that precisely characterizes total variation contamination models; and (3) construct anytime-valid p-values and contamination-adaptive test statistics—without requiring regularity conditions such as differentiability or boundedness. The framework rigorously controls Type-I error under arbitrary data-dependent stopping times and maintains high statistical power within the ε-neighborhood of the nominal model. Extensive simulations demonstrate substantial improvements over classical sequential tests—including SPRT and mixture-based methods—under various contamination regimes.

Technology Category

• Reasoning under Uncertainty: Sequential Decision Making
• Machine Learning: Adversarial Learning & Robustness
• Intelligent Robots: State Estimation

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

We propose an e-value based framework for testing composite nulls against composite alternatives when an $epsilon$ fraction of the data can be arbitrarily corrupted. Our tests are inherently sequential, being valid at arbitrary data-dependent stopping times, but they are new even for fixed sample sizes, giving type-I error control without any regularity conditions. We achieve this by modifying and extending a proposal by Huber (1965) in the point null versus point alternative case. Our test statistic is a nonnegative supermartingale under the null, even with a sequentially adaptive contamination model where the conditional distribution of each observation given the past data lies within an $epsilon$ (total variation) ball of the null. The test is powerful within an $epsilon$ ball of the alternative. As a consequence, one obtains anytime-valid p-values that enable continuous monitoring of the data, and adaptive stopping. We analyze the growth rate of our test supermartingale and demonstrate that as $epsilon o 0$, it approaches a certain Kullback-Leibler divergence between the null and alternative, which is the optimal non-robust growth rate. A key step is the derivation of a robust Reverse Information Projection (RIPr). Simulations validate the theory and demonstrate excellent practical performance.

Problem

• Develop robust tests for composite nulls and alternatives
• Control type-I error under data corruption
• Achieve optimal growth rate under contamination

Innovation

• E-value framework for robust composite testing
• Sequential supermartingale test statistic
• Robust Reverse Information Projection derivation