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

Peter D. Grünwald, Wouter M. Koolen

TL;DR

The paper addresses one-sided sequential testing and the challenge of maintaining type-I control under optional stopping when the null is expanded to $\delta\le\delta_0$. It shows that, provided there exists a sequence of sufficient statistics $T_n$ with the Monotone Likelihood Ratio Property, the corresponding likelihood-ratio process remains a test supermartingale for the enlarged null, even though the original conditional LR in the data $U_n$ may not be monotone. In particular, in the standard t-test setting, the statistic $T_n$ (the t-statistic) is sufficient and satisfies MLR, ensuring the LR process is a supermartingale; the framework extends to $\,\chi^2\,$-tests, linear regression, and label-agnostic Bernoulli experiments. The results yield anytime-valid, growth-rate optimal e-values against one-sided alternatives, with potential extensions to priors on the alternative and multivariate sufficiency, offering robust sequential testing tools for scale-invariant and nuisance-parameter-invariant problems.

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 $χ^2$-test and sequential linear regression with nuisance covariates.

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

TL;DR

The paper addresses one-sided sequential testing and the challenge of maintaining type-I control under optional stopping when the null is expanded to . It shows that, provided there exists a sequence of sufficient statistics with the Monotone Likelihood Ratio Property, the corresponding likelihood-ratio process remains a test supermartingale for the enlarged null, even though the original conditional LR in the data may not be monotone. In particular, in the standard t-test setting, the statistic (the t-statistic) is sufficient and satisfies MLR, ensuring the LR process is a supermartingale; the framework extends to -tests, linear regression, and label-agnostic Bernoulli experiments. The results yield anytime-valid, growth-rate optimal e-values against one-sided alternatives, with potential extensions to priors on the alternative and multivariate sufficiency, offering robust sequential testing tools for scale-invariant and nuisance-parameter-invariant problems.

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 -test and sequential linear regression with nuisance covariates.

Paper Structure

This paper contains 15 sections, 3 theorems, 31 equations.

Table of Contents

  1. The Problem
  2. Simple Null
  3. One-Sided Tests
  4. The Difficulty
  5. The Solution
  6. Further Discussion and Future Work
  7. Priors on Alternative and Growth-Rate Optimality Considerations
  8. Future Work
  9. Acknowledgements
  10. Proof for completeness
  11. Further Examples
  12. Variance of interest, translation nuisance ($\chi^2$-test)
  13. Linear Regression
  14. Label Agnostic Bernoulli
  15. No Free For All

Key Result

Proposition 2

Suppose that $T$ satisfies the MLR Property and fix $\delta \le \delta_0 \le \delta^+$ with $\delta, \delta_0,\delta^+ \in \Delta$. Then $\mathop{\mathrm{\mathbf E}}\nolimits_\delta[{p}^{T}_{\delta^+}(T)] \leq 1$, i.e. the likelihood ratio ${p}^{T}_{\delta^+}(T)$ is an e-variable for $\mathcal{H}_{\

Theorems & Definitions (9)

  • Definition 1: Monotone Likelihood Ratio Property
  • Proposition 2
  • Example 1: t-test
  • Lemma 3
  • Theorem 4
  • Example 2: t-test, Continued
  • proof : Proof of Lemma \ref{['lem:main']}
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Proposition \ref{['prop:monotoneb']}