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.

翻译：1976年，Lai针对未知方差$\sigma^2$的高斯分布均值$\mu$构建了一个非平凡的置信序列。值得注意的是，他同时使用了$\sigma$上的非正常（右哈尔）混合分布和$\mu$上的非正常（平坦）混合分布。本文对其构建细节进行了详细阐述，其中涉及广义不可积鞅和扩展的维尔不等式。尽管该方法确实给出了序贯t检验，但由于其鞅的非可积性，未能生成“e-过程”。针对同一设定，本文提出了两种新的e-过程及置信序列：一种是在缩减滤波下的检验鞅，另一种是规范数据滤波下的e-过程。这两种方法分别通过将Lai的平坦混合替换为高斯混合，以及将$\sigma$上的右哈尔混合替换为原假设下的最大似然估计（如通用推理中的做法）得到。我们还分析了所得置信序列的宽度，发现其与误差概率$\alpha$存在奇特的多项式依赖关系，并证明这种依赖不仅不可避免，而且（在通用推理中）甚至优于经典固定样本t检验。本文通过数值实验对比了多种方法（包括近期出现的次优方法）的性能差异。