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

• URL: http://arxiv.org/abs/2310.03722v5
• Date: Wed, 06 Nov 2024 22:27:19 GMT
• Title: Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance
• Authors: Hongjian Wang, Aaditya Ramdas,
• Abstract summary: In 1976, Lai constructed a nontrivial confidence sequence for the mean $mu$ of a Gaussian distribution with unknown variance. Here, we elaborate carefully on the details of his construction, which use generalized nonintegrable martingales and an extended Ville's inequality. We develop two new e-processes and confidence sequences for the same setting.
• Score: 30.14855064043107
• License:
• 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.

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