证明了Gaffke的置信下界在特定排序下对最大均值参数最优。
On the Order-Conditional Optimality of Gaffke's Bound
- 用概率语言重新构建经典置信下界理论，更易理解扩展。
- 在独立变量情形下，Gaffke界对最大边际均值达到最优。
- 适合统计推断与置信区间优化研究者阅读。
设 $X = (X_1, \ ldots, X_n)$ 为 $ R_+^n$ 上任意博雷尔概率分布的随机向量。本文重新审视对该分布某标量参数构造置信下界（LCB）的问题。通过纯概率论语言重构从Buehler开始的经典工作，形成更易理解且可扩展的框架。随后将该框架特化到分量独立的情形。在此设定下，证明了Gaffke的界对于其诱导的样本排序，在最大边际均值参数 $\max_{i \in [n]} \mathbb{E}_Q[X_i]$ 处达到Buehler最优；当 $X_i$ 独立同分布时，该参数退化为公共均值。即：任何按相同方式排序样本的合法置信下界，都无法在该参数上超越Gaffke界。
原文摘要 · Abstract (English)
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.
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