Testing the mean and variance by e-processes

Abstract

We address the problem of testing the conditional mean and conditional variance for nonstationary data. We build e -values and p -values for four types of nonparametric composite hypothesis with specified mean and variance as well as other conditions on the shape of the data-generating distribution. These shape conditions include symmetry, unimodality and their combination. Using the obtained e -values and p -values, we construct tests via e -processes, also known as testing by betting, as well as some tests based on combining p -values for comparison. Although we mainly focus on one-sided tests, the two-sided test for the mean is also studied. Simulation and empirical studies are conducted under a few settings, and they illustrate features of the methods based on e -processes.