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

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.