Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912107693342720 |
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| author | Wang, Hongjian Ramdas, Aaditya |
| author_facet | Wang, Hongjian Ramdas, Aaditya |
| contents | In 1976, Lai constructed a nontrivial confidence sequence for the mean $μ$ of a Gaussian distribution with unknown variance $σ^2$. Curiously, he employed both an improper (right Haar) mixture over $σ$ and an improper (flat) mixture over $μ$. 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 $σ$ 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 $α$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_03722 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance Wang, Hongjian Ramdas, Aaditya Statistics Theory Machine Learning Methodology In 1976, Lai constructed a nontrivial confidence sequence for the mean $μ$ of a Gaussian distribution with unknown variance $σ^2$. Curiously, he employed both an improper (right Haar) mixture over $σ$ and an improper (flat) mixture over $μ$. 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 $σ$ 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 $α$ 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. |
| title | Anytime-valid t-tests and confidence sequences for Gaussian means with unknown variance |
| topic | Statistics Theory Machine Learning Methodology |
| url | https://arxiv.org/abs/2310.03722 |