Testing mean and variance by e-processes

TL;DR

This paper develops e-value and p-value based tests for conditional mean and variance in non-stationary data, incorporating shape constraints like symmetry and unimodality, and demonstrates their effectiveness through simulations.

Contribution

It introduces novel e-process based testing methods for non-parametric hypotheses with shape constraints, expanding the toolkit for non-stationary data analysis.

Findings

E-process based tests effectively control error rates.

Shape constraints improve test power.

Simulations demonstrate practical applicability.

Abstract

We address the problem of testing conditional mean and conditional variance for non-stationary data. We build e-values and p-values for four types of non-parametric composite hypotheses 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.