Supermartingales for One-Sided Tests: Sufficient Monotone Likelihood Ratios are Sufficient

TL;DR

This paper demonstrates that for one-sided sequential tests, monotone likelihood ratios and sufficient statistics ensure control over false rejections in the opposite direction, with applications to t-tests, chi-squared tests, and regression.

Contribution

It establishes that monotone likelihood ratios are sufficient for controlling false rejections in one-sided sequential tests, extending the applicability of martingale methods.

Findings

Monotone likelihood ratios guarantee control of false positives in the opposite tail.

Applications include scale-invariant t-tests and sequential linear regression.

The approach unifies various statistical tests under a common martingale framework.

Abstract

The t-statistic is a widely-used scale-invariant statistic for testing the null hypothesis that the mean is zero. Martingale methods enable sequential testing with the t-statistic at every sample size, while controlling the probability of falsely rejecting the null. For one-sided sequential tests, which reject when the t-statistic is too positive, a natural question is whether they also control false rejection when the true mean is negative. We prove that this is the case using monotone likelihood ratios and sufficient statistics. We develop applications to the scale-invariant t-test, the location-invariant $\chi^2$-test and sequential linear regression with nuisance covariates.