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

2025/02/06 by Grünwald, Peter D., Koolen, Wouter M.
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2502.04208

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 χ²-test and sequential linear regression with nuisance covariates.

Related

• Test Martingales for bounded random variables
• A Sequential Test for Log-Concavity
• Anytime-valid sequential testing for elicitable functionals via supermartingales
• Rediscovering a little known fact about the t-test and the F-test: Algebraic, Geometric, Distributional and Graphical Considerations
• Admissible anytime-valid sequential inference must rely on nonnegative\n martingales