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。尽管这些实验在设计时未必以假设检验为目的，但研究者仍可能希望在实验完成后进行检验。本文旨在通过分析序贯实验的渐近性质，为开发其最优检验方法提供支持。我们的关键发现是：任意检验的渐近功效函数均可被一个极限实验中的检验所匹配，在该极限实验中，每个处理对应一个高斯过程，推断则针对这些过程的漂移项进行。这一结果具有重要含义，包括一个强大的充分性结论：无论序贯实验的类型如何，任一候选检验只需依赖一组固定的统计量。这些统计量包括实验结束时每个处理被采样的次数，以及每个处理的得分（针对参数模型）或有效影响函数（针对非参数模型）过程的终值。随后，我们在无偏性、α花费约束等不同限制条件下刻画了渐近最优检验。最后，我们将结果应用于三类关键的序贯实验：成本采样、组序贯试验以及赌臂实验，并展示了如何在这些场景中实施最优推断。