Optimal E-Values for Exponential Families: the Simple Case

TL;DR

This paper establishes a unifying condition for the existence of simple e-variables in testing composite hypotheses within exponential families, simplifying their computation and optimality analysis across various statistical tests.

Contribution

It provides a general theorem characterizing when simple e-variables exist for exponential family null hypotheses, extending previous specific cases to a broad class of tests.

Findings

Simple e-variables exist under a covariance matrix condition.

Applicable to linear regression, k-sample, Gaussian location and scale tests.

Enables easy computation and optimality of e-variables in complex testing scenarios.

Abstract

We provide a general condition under which e-variables in the form of a simple-vs.-simple likelihood ratio exist when the null hypothesis is a composite, multivariate exponential family. Such `simple' e-variables are easy to compute and expected-log-optimal with respect to any stopping time. Simple e-variables were previously only known to exist in quite specific settings, but we offer a unifying theorem on their existence for testing exponential families. We start with a simple alternative $Q$ and a regular exponential family null. Together these induce a second exponential family ${\cal Q}$ containing $Q$, with the same sufficient statistic as the null. Our theorem shows that simple e-variables exist whenever the covariance matrices of ${\cal Q}$ and the null are in a certain relation. A prime example in which this relation holds is testing whether a parameter in a linear regression is 0. Other examples include some $k$-sample tests, Gaussian location- and scale tests, and tests for more general classes of natural exponential families. While in all these examples, the implicit composite alternative is also an exponential family, in general this is not required.