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

arXiv: https://arxiv.org/abs/2310.03722 · HTML: https://mlsanity.com/paper/2310.03722/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 5 Oct 2023.

## Builds on (most cited first, 18 of 18)

- [On the Complexity of Best Arm Identification in Multi-Armed Bandit Models](https://mlsanity.com/paper/1407.4443/) (2014)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Safe Testing](https://mlsanity.com/paper/1906.07801/) (2019)
- [Explore First, Exploit Next: The True Shape of Regret in Bandit Problems](https://mlsanity.com/paper/1602.07182/) (2016)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Testing exchangeability: fork-convexity, supermartingales, and e-processes](https://mlsanity.com/paper/2102.00630/) (2021)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [Sequential estimation of quantiles with applications to A/B-testing and best-arm identification](https://mlsanity.com/paper/1906.09712/) (2019)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities](https://mlsanity.com/paper/2304.02611/) (2023)
- [Dealing with Unknown Variances in Best-Arm Identification](https://mlsanity.com/paper/2210.00974/) (2022)
- [A composite generalization of Ville's martingale theorem using e-processes](https://mlsanity.com/paper/2203.04485/) (2022)
- [A unified framework for bandit multiple testing](https://mlsanity.com/paper/2107.07322/) (2021)
- [E-Statistics, Group Invariance and Anytime Valid Testing](https://mlsanity.com/paper/2208.07610/) (2022)
- [Anytime-Valid Linear Models and Regression Adjusted Causal Inference in Randomized Experiments](https://mlsanity.com/paper/2210.08589/) (2022)
- [Efficiency of nonparametric e-tests](https://mlsanity.com/paper/2208.08925/) (2022)
- [The extended Ville's inequality for nonintegrable nonnegative supermartingales](https://mlsanity.com/paper/2304.01163/) (2023)

## Built on by (most cited first, 2 of 2)

- [E-valuator: Reliable Agent Verifiers with Sequential Hypothesis Testing](https://mlsanity.com/paper/2512.03109/) (2025)
- [High-probability minimax lower bounds](https://mlsanity.com/paper/2406.13447/) (2024)

## Abstract (arXiv)

In 1976, Lai constructed a nontrivial confidence sequence for the mean μ of a Gaussian distribution with unknown variance σ2. 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.