A Sequential Test for Log-Concavity

TL;DR

This paper investigates the challenge of testing for log-concavity in i.i.d. data distributions, proving the non-existence of certain martingale tests and proposing a universal inference-based e-process that is consistent and powerful.

Contribution

It introduces the first sequential test for log-concavity using universal inference, overcoming the non-existence of test martingales for this nonparametric class.

Findings

Proves no nontrivial test martingales exist for log-concave distributions.

Constructs a new e-process-based test that is consistent and level-lpha.

Demonstrates the test's power against Hellinger alternatives.

Abstract

On observing a sequence of i.i.d.\ data with distribution $P$ on $\mathbb{R}^d$, we ask the question of how one can test the null hypothesis that $P$ has a log-concave density. This paper proves one interesting negative and positive result: the non-existence of test (super)martingales, and the consistency of universal inference. To elaborate, the set of log-concave distributions $\mathcal{L}$ is a nonparametric class, which contains the set $\mathcal G$ of all possible Gaussians with any mean and covariance. Developing further the recent geometric concept of fork-convexity, we first prove that there do no exist any nontrivial test martingales or test supermartingales for $\mathcal G$ (a process that is simultaneously a nonnegative supermartingale for every distribution in $\mathcal G$), and hence also for its superset $\mathcal{L}$. Due to this negative result, we turn our attention to constructing an e-process -- a process whose expectation at any stopping time is at most one, under any distribution in $\mathcal{L}$ -- which yields a level-$\alpha$ test by simply thresholding at $1/\alpha$. We take the approach of universal inference, which avoids intractable likelihood asymptotics by taking the ratio of a nonanticipating likelihood over alternatives against the maximum likelihood under the null. Despite its conservatism, we show that the resulting test is consistent (power one), and derive its power against Hellinger alternatives. To the best of our knowledge, there is no other e-process or sequential test for $\mathcal{L}$.