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

arXiv: https://arxiv.org/abs/2607.18661 · HTML: https://mlsanity.com/paper/2607.18661/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2026.

## Built on by (most cited first, 1 of 1)

- [On the Order-Conditional Optimality of Gaffke's Bound](https://mlsanity.com/paper/2607.22971/) (2026)

## Abstract (arXiv)

Given observations mathbf x=(x1,dots,xn), Gaffke (2005) defined [ Kn(mathbf x)=mathbbPmathbf D!leftsumi=1n xiDi≤ 1right, qquad (D0,D1,ldots,Dn)~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∈[0,∞)n and every elementary symmetric polynomial ek, ( Kn(mathbf x)ek(mathbf x)≤ nchoose 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 K2. A neutral-face extension proves inadmissibility of Kn for every n≥2. If one independent uniform random variable is allowed, there is an even simpler full-dimensional improvement: on the upper orthant, where Kn(mathbf x)=1/prodi xi, replace it by U/prodi xi. The equal-tail Gaffke confidence interval In is nevertheless first-order asymptotically efficient: for iid observations on [0,1] with unknown variance σ2>0, [ √ n,operatornameWidth(In)longrightarrow 2σ z1-α/2qquadalmost 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.