General Behaviour of P-Values Under the Null and Alternative

TL;DR

This paper investigates the behavior of p-values under null and alternative hypotheses, revealing that common assumptions can be misleading and proposing corrected tests for more accurate inference.

Contribution

It characterizes the distribution of p-values using higher order asymptotics and introduces corrected tests that outperform traditional methods in certain scenarios.

Findings

Common beliefs about p-value distributions can be misleading.

Corrected tests improve accuracy over first order tests.

Higher order asymptotics provide better understanding of p-value behavior.

Abstract

Hypothesis testing results often rely on simple, yet important assumptions about the behaviour of the distribution of p-values under the null and the alternative. We examine tests for one dimensional parameters of interest that converge to a normal distribution, possibly in the presence of nuisance parameters, and characterize the distribution of the p-values using techniques from the higher order asymptotics literature. We show that commonly held beliefs regarding the distribution of p-values are misleading when the variance and location of the test statistic are not well-calibrated or when the higher order cumulants of the test statistic are not negligible. Corrected tests are proposed and are shown to perform better than their first order counterparts in certain settings.