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

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  • Grünwald, Peter
  • Koolen, Wouter M.

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.

Suggested Citation

  • Grünwald, Peter & Koolen, Wouter M., 2026. "Supermartingales for one-sided tests: Sufficient monotone likelihood ratios are sufficient," Statistics & Probability Letters, Elsevier, vol. 229(C).
    • Handle: RePEc:eee:stapro:v:229:y:2026:i:c:s0167715225002196
    DOI: 10.1016/j.spl.2025.110574
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    References listed on IDEAS

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    1. Jahar Bhowmik & Maxwell King, 2007. "Maximal invariant likelihood based testing of semi-linear models," Statistical Papers, Springer, vol. 48(3), pages 357-383, September.
    2. Larsson, Martin & Ramdas, Aaditya & Ruf, Johannes, 2025. "The numeraire e-variable and reverse information projection," LSE Research Online Documents on Economics 126527, London School of Economics and Political Science, LSE Library.
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