发表机构
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究推导随机向量标量参数下置信界问题，重塑经典工作形成概率框架并用于\(X\)分量独立情况，证明加夫克界对最大边际均值参数的序条件最优，无其他同序有效LCB能超越。
AI 中文摘要
设\(X=(X_1,\ldots,X_n)\)是来自\(\mathbb{R}_+^n\)上任意波莱尔概率律的随机向量。我们重新审视推导该概率律标量参数的下置信界（LCB）问题。我们用纯概率术语重塑始于比勒的经典工作，以形成更易理解和可扩展的框架。然后将该框架专门应用于\(X\)的分量独立的情况。在此背景下，我们证明加夫克界对于它所诱导的关于最大边际均值参数\(\max_{i\in[n]}E_Q[X_i]\)的序是比勒最优的，当\(X_i\)独立同分布时该参数简化为共同均值。也就是说，没有其他以与加夫克界相同方式对样本排序的有效LCB能在该参数方面优于它。
英文摘要
Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensible framework. We then specialize the framework to the case where the components of $X$ are independent. In this context, we prove that Gaffke's bound is Buehler optimal for the order that it induces with respect to the maximum marginal mean parameter: $max_{i \in [n]} E_Q[X_i]$, which reduces to the common mean when the $X_i$ are independent and identically distributed. That is to say, no other valid LCB that orders samples in the same way as Gaffke's bound can improve on it with respect to this parameter.
