Optimal Anytime-Valid Tests for Composite Nulls

TL;DR

This paper develops optimal anytime-valid tests for composite null hypotheses, achieving the theoretical lower bounds on expected sample size, using universal e-processes and saddle-point representations, with practical considerations for implementation.

Contribution

It introduces a constructive method for optimal tests based on e-processes and saddle-point analysis, applicable to finite and infinite sample spaces, extending prior theoretical bounds.

Findings

Optimal tests match the lower bound as significance level decreases.

Universal e-processes based on empirical saddle-point solutions are optimal.

Conditions for optimality are verified for H"older smooth density models.

Abstract

We consider the problem of designing optimal level-$\alpha$ power-one tests for composite nulls. Given a parameter $\alpha \in (0,1)$ and a stream of $\mathcal{X}$-valued observations $\{X_n: n \geq 1\} \overset{i.i.d.}{\sim} P$, the goal is to design a level-$\alpha$ power-one test $\tau_\alpha$ for the null $H_0: P \in \mathcal{P}_0 \subset \mathcal{P}(\mathcal{X})$. Prior works have shown that any such $\tau_\alpha$ must satisfy $\mathbb{E}_P[\tau_\alpha] \geq \tfrac{\log(1/\alpha)}{\gamma^*(P, \mathcal{P}_0)}$, where $\gamma^*(P, \mathcal{P}_0)$ is the so-called $\mathrm{KL}_{\inf}$ or minimum divergence of $P$ to the null class. In this paper, our objective is to develop and analyze constructive schemes that match this lower bound as $\alpha \downarrow 0$. We first consider the finite-alphabet case~($|\mathcal{X}| = m < \infty$), and show that a test based on \emph{universal} $e$-process~(formed by the ratio of a universal predictor and the running null MLE) is optimal in the above sense. The proof relies on a Donsker-Varadhan~(DV) based saddle-point representation of $\mathrm{KL}_{\inf}$, and an application of Sion's minimax theorem. This characterization motivates a general method for arbitrary $\mathcal{X}$: construct an $e$-process based on the empirical solutions to the saddle-point representation over a sufficiently rich class of test functions. We give sufficient conditions for the optimality of this test for compact convex nulls, and verify them for H\"older smooth density models. We end the paper with a discussion on the computational aspects of implementing our proposed tests in some practical settings.