Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance

TL;DR

This paper develops new anytime-valid confidence sequences and e-processes for Gaussian means with unknown variance, improving upon Lai's classical construction and analyzing their widths and properties.

Contribution

It introduces two novel e-processes for Gaussian mean inference with unknown variance, using mixture strategies and maximum likelihood, extending Lai's work with rigorous analysis.

Findings

New confidence sequences have polynomial dependence on error probability.

The proposed methods outperform some recent suboptimal approaches.

Width analysis shows advantages over classical fixed-sample t-tests.

Abstract

In 1976, Lai constructed a nontrivial confidence sequence for the mean $\mu$ of a Gaussian distribution with unknown variance $\sigma^2$. Curiously, he employed both an improper (right Haar) mixture over $\sigma$ and an improper (flat) mixture over $\mu$. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. While this does yield a sequential t-test, it does not yield an "e-process" (due to the nonintegrability of his martingale). In this paper, we develop two new e-processes and confidence sequences for the same setting: one is a test martingale in a reduced filtration, while the other is an e-process in the canonical data filtration. These are respectively obtained by swapping Lai's flat mixture for a Gaussian mixture, and swapping the right Haar mixture over $\sigma$ with the maximum likelihood estimate under the null, as done in universal inference. We also analyze the width of resulting confidence sequences, which have a curious polynomial dependence on the error probability $\alpha$ that we prove to be not only unavoidable, but (for universal inference) even better than the classical fixed-sample t-test. Numerical experiments are provided along the way to compare and contrast the various approaches, including some recent suboptimal ones.