Sequential testing, always-valid $p$-values, and confidence sequences promise flexible statistical inference and on-the-fly decision making. However, unlike fixed-$n$ inference based on asymptotic normality, existing sequential tests either make parametric assumptions and end up under-covering/over-rejecting when these fail or use non-parametric but conservative concentration inequalities and end up over-covering/under-rejecting. To circumvent these issues, we sidestep exact at-least-$\alpha$ coverage and focus on asymptotic calibration and asymptotic optimality. That is, we seek sequential tests whose probability of \emph{ever} rejecting a true hypothesis approaches $\alpha$ and whose expected time to reject a false hypothesis approaches a lower bound on all such asymptotically calibrated tests, both "approaches" occurring under an appropriate limit. We permit observations to be both non-parametric and dependent and focus on testing whether the observations form a martingale difference sequence. We propose the universal sequential probability ratio test (uSPRT), a slight modification to the normal-mixture sequential probability ratio test, where we add a burn-in period and adjust thresholds accordingly. We show that even in this very general setting, the uSPRT is asymptotically optimal under mild generic conditions. We apply the results to stabilized estimating equations to test means, treatment effects, {\etc} Our results also provide corresponding guarantees for the implied confidence sequences. Numerical simulations verify our guarantees and the benefits of the uSPRT over alternatives.

翻译：序贯检验、始终有效的$p$值和置信序列为灵活的统计推断和动态决策提供了可能。然而，与基于渐近正态性的固定样本量推断不同，现有序贯检验要么依赖于参数假设（但当假设不成立时会导致覆盖不足/过度拒绝），要么采用非参数但保守的集中不等式（从而导致过度覆盖/拒绝不足）。为解决这些问题，我们放弃精确的至少$\alpha$覆盖率，转而专注于渐近校准和渐近最优性。具体而言，我们寻求这样一类序贯检验：其在原假设为真时“曾经”拒绝原假设的概率趋近于$\alpha$，且拒绝错误假设的期望时间趋近于所有渐近校准检验的下界——这两个“趋近”均发生在适当极限下。我们允许观测值具有非参数性和依赖性，并重点检验观测值是否构成鞅差序列。我们提出通用序贯概率比检验（uSPRT），这是对正态混合序贯概率比检验的轻微改进：在其中加入预热期并相应调整阈值。我们证明，即使在这种非常一般的设定下，uSPRT在温和的通用条件下也是渐近最优的。我们将该结果应用于稳定估计方程以检验均值、处理效应等。我们的结果也为隐含的置信序列提供了相应的保证。数值模拟验证了我们的保证以及uSPRT相较于其他方法的优势。