基于有界数据均值投注的高斯高效检验
Gaussian-efficient testing by betting on the mean of bounded data
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中文总结 AI 辅助
该研究针对有界数据均值提出基于新型投注策略的高斯高效非渐近置信区间，兼具鞅依赖下有限样本有效性、计算简便性，在模拟中表现优于STaR-Bets等方法，还扩展至无放回抽样场景。
中文摘要 AI 辅助
给定取值范围为[0,1]的随机变量X₁,…,Xₙ，满足对所有i，条件期望E[X_i | X₁,…,X_{i-1}]=μ，我们提出一种新的非渐近置信区间，该区间通过反转由新型投注策略生成的终端e-值得到。当数据独立同分布（iid）时，其极限宽度与中心极限定理的结果匹配（即“高斯高效”），最终超越了先前投注区间的低效极限。我们的主要概念进展在于设计投注分数，使其跟踪极限高斯实验中最强大终端检验的条件拒绝概率。当一个可预测方差估计量在候选均值间共享时，确定性反转可得到适用于所有数据序列的区间，且其两个端点易于查找，通过外部随机化可进一步减小宽度。模拟实验显示，我们的方法能生成迄今最紧凑的区间：在所有测试分布及所有足够大的n下，其确定性版本优于STaR-Bets，且与Gaffke方法具有竞争力；随机化改进版本则优于两者。该方法兼具鞅依赖下的有限样本有效性、端点计算简便性、针对iid数据的高斯高效推断以及优异的经验性能，我们还将该构造及其效率理论扩展至无放回抽样场景，在该场景下同样达到了顶尖的经验性能。
英文摘要
Given $[0,1]$-valued random variables $X_1,\dots,X_n$ such that $\mathbb{E}[X_i | X_1,\dots,X_{i-1}]= μ$ for all $i$, we propose a new nonasymptotic confidence interval for $μ$ that is obtained by inverting terminal e-values generated by a novel betting strategy. When the data are iid, its limiting width matches that of the central limit theorem (``Gaussian-efficient''), finally surpassing the inefficient limits of previous betting intervals. Our main conceptual advance involves designing betting fractions that track the conditional rejection probability of the most powerful terminal test in a limiting Gaussian experiment. When one predictable variance estimator is shared across candidate means, the deterministic inversion is an interval for every data sequence and its two endpoints can be found easily. The width can be improved further with external randomization. In simulations, our method yields the tightest intervals to date; for every distribution tested and all sufficiently large $n$, our deterministic version beats STaR-Bets and is competitive with Gaffke, while the randomized improvement beats both. It thus combines finite-sample validity under martingale dependence, easy endpoint computation, Gaussian-efficient inference for iid data, and excellent empirical performance. We also extend the construction and its efficiency theory to sampling without replacement, where it again achieves state-of-the-art empirical performance.
