加夫克关于有界数据均值的置信区间是不可容许的但渐近有效
Gaffke's confidence interval for the mean of bounded data is inadmissible but asymptotically efficient
AI总结：
研究加夫克关于有界数据均值的置信区间，通过分析其检验和区间的有限样本与大样本性质，发现该区间不可容许但渐近有效，还通过模拟表明在多种有界均值区间中它是最短的。
AI中文摘要：
给定观测值\(\mathbf x=(x_1,\dots,x_n)\)，加夫克（2005年）定义了\[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \]并推测当输入为独立e值时它是一个p值。最近，弗拉西斯和托马斯（2026年）证明了这一推测。对\([0,1]\)中的观测值进行检验反演得到了利尔内德 - 米勒和托马斯（2020年）研究的置信区间，对于伯努利数据它简化为克洛普 - 皮尔逊区间。我们给出了加夫克检验和区间的有限样本和大样本分析。首先，对于每个\(\mathbf x\in[0,\infty)^n\)和每个初等对称多项式\(e_k\)，\( K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}\)，所以加夫克p值从不大于明等人（2026年）的SymPol p值。然而，加夫克的p值是不可容许的。对于\(n = 2\)，我们构造了一个有效规则，在混合配置上严格更小，并且是支配\(K_2\)的唯一可容许规则。一个中性面扩展证明了对于每个\(n\ge2\)，\(K_n\)是不可容许的。如果允许一个独立均匀随机变量，有一个更简单的全维改进：在上正交卦限，其中\(K_n(\mathbf x)=1/\prod_i x_i\)，将其替换为\(U/\prod_i x_i\)。等尾加夫克置信区间\(I_n\)仍然是一阶渐近有效的：对于\([0,1]\)上具有未知方差\(\sigma^2>0\)的独立同分布观测值，\[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2\sigma z_{1-\alpha/2}\qquad\text{几乎必然}。\]我们的模拟还发现，在考虑的各种有界均值区间中，加夫克区间是最短的，包括与具有相同一阶高斯目标的最近经验贝里 - 埃森程序的比较。
英文摘要：
Given observations $\mathbf x=(x_1,\dots,x_n)$, Gaffke (2005) defined \[ K_n(\mathbf x)=\mathbb{P}_{\mathbf D}\!\left\{\sum_{i=1}^n x_iD_i\le 1\right\}, \qquad (D_0,D_1,\ldots,D_n)\sim\mathrm{Dirichlet}(1,\ldots,1), \] and conjectured that it is a $p$-value whenever the inputs are independent e-values. Recently, Vlassis and Thomas (2026) proved this conjecture. Inverting the tests for observations in $[0,1]$ gives the confidence interval studied by Learned-Miller and Thomas (2020), which reduces to Clopper--Pearson for Bernoulli data. We give a finite- and large-sample account of Gaffke's test and interval. First, for every $\mathbf x\in[0,\infty)^n$ and every elementary symmetric polynomial $e_k$, \( K_n(\mathbf x)e_k(\mathbf x)\le {n\choose k}, \) so the Gaffke $p$-value never larger than the SymPol $p$-value of Ming et al. (2026). However, Gaffke's p-value is inadmissible. For $n=2$, we construct a valid rule that is strictly smaller on mixed configurations and is the unique admissible rule that dominates $K_2$. A neutral-face extension proves inadmissibility of $K_n$ for every $n\ge2$. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where $K_n(\mathbf x)=1/\prod_i x_i$, replace it by $U/\prod_i x_i$. The equal-tail Gaffke confidence interval $I_n$ is nevertheless first-order asymptotically efficient: for iid observations on $[0,1]$ with unknown variance $σ^2>0$, \[ \sqrt n\,\operatorname{Width}(I_n)\longrightarrow 2σz_{1-α/2}\qquad\text{almost surely}. \] Our simulations also find that, among a variety of bounded-mean intervals considered, the Gaffke interval is the shortest, including comparisons with a recent empirical Berry--Esseen procedure having the same first-order Gaussian target.
