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针对方差$\sigma^2$未知的高斯分布均值$\mu$构建了一个非平凡的置信序列。值得注意的是，他同时采用了关于$\sigma$的非正常（右哈尔）混合与关于$\mu$的非正常（平坦）混合。本文详细阐述其构造细节，其中运用了广义不可积鞅与扩展的Ville不等式。虽然该方法确实产生了序贯t检验，但由于其鞅的不可积性，并未产生“e-过程”。本文针对同一设定提出了两种新的e-过程与置信序列：其一是简化滤波中的检验鞅，其二则是规范数据滤波中的e-过程。它们分别通过以下方式获得：将Lai的平坦混合替换为高斯混合；以及将$\sigma$的右哈尔混合替换为在原假设下采用极大似然估计的方法（如通用推断所示）。我们还分析了所得置信序列的宽度，其与误差概率$\alpha$存在奇特的多项式依赖关系。我们证明这种依赖不仅是不可避免的，而且（对于通用推断而言）甚至优于经典固定样本t检验。文中通过数值实验比较和对比了各类方法（包括近期一些次优方法）的特性。