Sequential tests and their implied confidence sequences, which are valid at arbitrary stopping times, promise flexible statistical inference and on-the-fly decision making. However, strong guarantees are limited to parametric sequential tests that under-cover in practice or concentration-bound-based sequences that over-cover and have suboptimal rejection times. In this work, we consider \cite{robbins1970boundary}'s delayed-start normal-mixture sequential probability ratio tests, and we provide the first asymptotic type-I-error and expected-rejection-time guarantees under general non-parametric data generating processes, where the asymptotics are indexed by the test's burn-in time. The type-I-error results primarily leverage a martingale strong invariance principle and establish that these tests (and their implied confidence sequences) have type-I error rates approaching a desired $\alpha$-level. The expected-rejection-time results primarily leverage an identity inspired by It\^o's lemma and imply that, in certain asymptotic regimes, the expected rejection time approaches the minimum possible among $\alpha$-level tests. We show how to apply our results to sequential inference on parameters defined by estimating equations, such as average treatment effects. Together, our results establish these (ostensibly parametric) tests as general-purpose, non-parametric, and near-optimal. We illustrate this via numerical experiments.

翻译：序贯检验及其隐含的置信序列（在任意停时下均有效）提供了灵活的统计推断与实时决策能力。然而，现有方法存在显著局限：参数化序贯检验虽具强理论保证却在实际中覆盖不足，而基于浓度界的置信序列则过度覆盖且拒绝时间次优。本文研究 \cite{robbins1970boundary} 提出的延迟启动正态混合序贯概率比检验，首次在一般非参数数据生成过程下给出了其渐近第一类错误率和期望拒绝时间保证（渐近性以检验的预热时间为索引）。第一类错误率结果主要利用鞅强不变原理，证明此类检验（及其隐含的置信序列）的错误率趋近于预设的 $\alpha$ 水平。期望拒绝时间结果主要基于受伊藤引理启发的恒等式，在特定渐近框架下表明期望拒绝时间趋近于 $\alpha$ 水平检验的最小可能值。我们展示了如何将结果应用于由估计方程定义的参数（如平均处理效应）的序贯推断。综合而言，本文证明这些表面上的参数化检验实则具有通用性、非参数性与近最优性，并通过数值实验加以验证。