发表机构
University of California, Los Angeles（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了在联合无原子性和绝对连续性假设下，精确且枢轴的e变量的期望对数上确界除以样本量收敛于最小KL散度，解决了Zhang等人提出的开放问题。
AI 中文摘要
考虑使用n个独立同分布观测值检验有限复合零假设$\cP={P_1,\ldots,P_L}$与简单备择假设$Q$。设$\ell_n$表示在所有在每个$P_i$下精确且在各零假设间枢轴的e变量上，期望对数e值的上确界。Zhang、Ramdas和Wang（2024）证明了$\ell_n$是超可加的，并询问在它们的联合无原子性和绝对连续性假设下，$\ell_n/n$是否收敛到上界$\min_i D(Q|P_i)$。我们证明它确实收敛。证明利用了Zhang、Ramdas和Wang（2024）以及Farooq、Fritz、Haapasalo和Tomamichel（2024）发展的技术。
英文摘要
Consider testing a finite composite null $\cP={P_1,\ldots,P_L}$ against a simple alternative $Q$ using $n$ i.i.d. observations. Let $\ell_n$ denote the supremum of the expected log e-value over e-variables that are exact under every $P_i$ and pivotal across the nulls. Zhang, Ramdas, and Wang (2024) showed that $\ell_n$ is superadditive and asked whether $\ell_n/n$ converges to the upper bound $\min_i D(Q|P_i)$ under their joint atomlessness and absolute continuity assumptions. We prove that it does. The proof draws on techniques developed by Zhang, Ramdas, and Wang (2024) and by Farooq, Fritz, Haapasalo, and Tomamichel (2024).
