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

TL;DR

This paper explores growth-optimal e-variables for multivariate hypotheses, extending the Csiszár-Sanov-Chernoff theorem to cases where the alternative hypothesis set surrounds the null, providing new theoretical insights.

Contribution

It introduces a novel extension of the Csiszár-Sanov-Chernoff theorem for multivariate hypotheses where the alternative set surrounds the null, and relates growth-optimal e-variables to these bounds.

Findings

Reformulation of known Csiszár-Sanov-Chernoff bounds for convex sets.

New results for cases where the alternative surrounds the null hypothesis.

Connections between growth-optimal e-variables and multivariate hypothesis testing.

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).