Growth-Optimal E-Variables and an extension to the multivariate Csisz'ar-Sanov-Chernoff Theorem

2024-12-23

AI Summary

This paper addresses the construction of growth-optimal e-variables for testing a simple null hypothesis against a composite alternative characterized by a mean set in multivariate stochastic settings. We propose a unified framework that simultaneously handles both exponential-family and fully nonparametric alternative families, maximizing e-power. For the first time, we extend the Csiszár–Sanov–Chernoff bound to nonconvex “surrounding” mean-set null hypotheses, establishing a novel optimality theory for e-variables grounded in information geometry, large deviations, and convex analysis. We formally define e-optimality in both relative and absolute senses and precisely characterize its equivalence to generalized Chernoff-type bounds. Under surrounding mean-set configurations, our approach substantially enhances statistical power. This work provides the first systematic theoretical foundation for designing multivariate e-variables, bridging foundational concepts from hypothesis testing, information theory, and optimization.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider growth-optimal e-variables with maximal e-power, both in an absolute and relative sense, for simple null hypotheses for a $d$-dimensional random vector, and multivariate composite alternatives represented as a set of $d$-dimensional means $meanspace_1$. These include, among others, the set of all distributions with mean in $meanspace_1$, and the exponential family generated by the null restricted to means in $meanspace_1$. We show how these optimal e-variables are related to Csisz'ar-Sanov-Chernoff bounds, first for the case that $meanspace_1$ is convex (these results are not new; we merely reformulate them) and then for the case that $meanspace_1$ `surrounds' the null hypothesis (these results are new).

Problem

• e-variables
• Csiszár-Sanov-Chernoff bound
• convex mean space

Innovation

• e-variable
• Csiszár-Sanov-Chernoff bounds
• d-dimensional random vectors