Optimal tests following sequential experiments

TL;DR

This paper develops a theoretical framework for optimal hypothesis testing after sequential experiments, showing that tests can be based on simple statistics and characterizing their asymptotic properties across different experimental designs.

Contribution

It introduces a limit experiment approach for sequential tests, providing a unified method to derive asymptotically optimal tests under various constraints.

Findings

Asymptotic power functions can be matched by tests in a Gaussian process limit experiment.

Optimal tests depend only on treatment sample counts and final scores or influence functions.

The framework applies to costly sampling, group trials, and bandit experiments.

Abstract

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.