Gaffke's confidence interval for the mean of bounded data is inadmissible but asymptotically efficient

2026-07-20

AI Summary

This study investigates the admissibility and efficiency of Gaffke’s confidence intervals for estimating the mean of bounded data. Focusing on the p-value derived from a Dirichlet distribution and its associated confidence interval, we demonstrate that this p-value is inadmissible and construct an improved decision rule that strictly dominates it for all sample sizes \( n \geq 2 \). Although the resulting confidence interval is also inadmissible, it achieves first-order asymptotic efficiency. Our theoretical analysis integrates symmetric polynomial inequalities, variable transformations, confidence interval inversion, and asymptotic methods. Simulation studies confirm that Gaffke’s interval consistently yields the shortest width across various bounded-mean settings and almost surely attains first-order Gaussian efficiency.

Abstract

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.

Problem

• confidence interval
• bounded data
• admissibility
• p-value
• asymptotic efficiency

Innovation

• Gaffke confidence interval
• inadmissibility
• asymptotic efficiency
• elementary symmetric polynomial
• p-value dominance