Recent years have seen tremendous advances in the theory and application of sequential experiments. While these experiments are not always designed with hypothesis testing in mind, researchers may still be interested in performing tests after the experiment is completed. The purpose of this paper is to aid in the development of optimal tests for sequential experiments by analyzing their asymptotic properties. Our key finding is that the asymptotic power function of any test can be matched by a test in a limit experiment where a Gaussian process is observed for each treatment, and inference is made for the drifts of these processes. This result has important implications, including a powerful sufficiency result: any candidate test only needs to rely on a fixed set of statistics, regardless of the type of sequential experiment. These statistics are the number of times each treatment has been sampled by the end of the experiment, along with final value of the score (for parametric models) or efficient influence function (for non-parametric models) process for each treatment. We then characterize asymptotically optimal tests under various restrictions such as unbiasedness, \alpha-spending constraints etc. Finally, we apply our our results to three key classes of sequential experiments: costly sampling, group sequential trials, and bandit experiments, and show how optimal inference can be conducted in these scenarios.

翻译：近年来，序贯实验的理论与应用取得了巨大进展。尽管这类实验在设计时未必以假设检验为目标，但研究者仍可能希望实验完成后进行检验。本文旨在通过分析其渐近性质，为序贯实验开发最优检验方法。核心发现是：任何检验的渐近功效函数均可在极限实验中通过某个检验实现，该极限实验中对每个处理观测一个高斯过程，并基于这些过程的漂移项进行推断。这一结果具有重要启示，包括一个强有力的充分性结论：无论序贯实验类型如何，任何候选检验仅需依赖固定的统计量集合。这些统计量包括实验结束时各处理被采样的次数，以及每个处理的得分（参数模型）或有效影响函数（非参数模型）过程的最终值。我们随后在无偏性、α-支出约束等条件下刻画了渐近最优检验。最后，将研究成果应用于三类关键序贯实验：昂贵采样实验、成组序贯试验和赌博机实验，并展示了如何在这些场景中进行最优推断。