E-values and sequential power-one tests for monotonicity and unimodality

TL;DR

This paper introduces e-values and e-processes for testing monotonicity and unimodality of distributions, providing powerful, consistent tests and estimators with theoretical guarantees.

Contribution

It develops the first e-value based tests for monotonicity and unimodality, characterizes all valid e-values, and extends results to continuous variables.

Findings

Tests achieve power one under any non-null distribution with i.i.d. data.

Consistent set-valued mode estimators eventually identify the true modes.

Characterization of all valid e-values and the most powerful e-value for fixed alternatives.

Abstract

We develop e-values and e-processes testing the null hypothesis that a distribution over nonnegative integers is monotone, and that a distribution over integers is unimodal given a certain mode. Our e-processes lead to tests of power one under any non-null distribution with a sequence of i.i.d. observations, and consistent set-valued mode estimators that eventually equal the true set of modes. Additionally, we characterize the set of all e-values, and therefore the set of all valid tests, with one monotone and unimodal observation, as well as the most powerful e-value for a fixed alternative. We then show that many of our results can be generalized to continuous random variables, relating them to the existing results in the shape-constrained inference literature.