Supermartingales for one-sided tests: Sufficient monotone likelihood ratios are sufficient

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 χ 2 -test and sequential linear regression with nuisance covariates. • We revisit the sequential one-sided t-test of positive effect size against zero effect size. • We show that it is in fact a supermartingale against any negative effect size. • The monotone likelihood ratio property is the key.