Ranking Experiments under Sequential Sampling

Abstract

We compare statistical experiments when observations are inexpensive and can be acquired sequentially until the decision maker chooses to stop. We introduce two orders. Small-cost decision dominance asks which of two equally priced experiments is eventually preferred in every decision problem as the per-observation cost vanishes; large-budget stopping dominance asks which experiment can reproduce every terminal experiment attainable from the other under all sufficiently large expected-sample budgets. Our main result shows that, for generic pairs, the two orders coincide and are both characterized by strict dominance of every pairwise Kullback--Leibler divergence. The key step is a uniform exact-conversion theorem: any finite-output stopping policy based on one experiment can be reproduced exactly using another, with first-order expected-sample requirements determined by pairwise KL rates and a square-root remainder that is uniform over policies.