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能在任意停止时间保持有效，从而支持灵活的统计推断与即时决策。然而，强保证仅局限于实践中覆盖不足的参数化序贯检验，或基于集中界、过度覆盖且剔除时间次优的序列。本文针对《罗宾斯1970边界》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中建立了渐近第一类错误率与预期剔除时间保证，其中渐近性以检验的预热时间为索引。第一类错误结果主要利用鞅强不变性原理，证明这些检验（及其隐含的置信序列）的第一类错误率趋近于预设的α水平。预期剔除时间结果主要借助伊藤引理启发的恒等式，表明在特定渐近框架下，预期剔除时间趋近于α水平检验可能的最小值。我们展示了如何将结果应用于估计方程（如平均处理效应）所定义参数的序贯推断。综合而言，本文结果确立了这些（看似参数化的）检验具有通用性、非参数性与近乎最优性，并通过数值实验加以验证。