2026/07/21 by Jiahao Ming, Aaditya Ramdas, Yi Shen +2 · 3 citations
#math.ST #cs.IT #eess.SP #math.IT #math.PR #stat.ME #stat.TH
Given observations \mathbf x=(x1,…,xn), Gaffke (2005) defined Kn(\mathbf x)=ℙ\mathbf D \∑i=1n xiDi≤ 1\, (D0,D1,…,Dn)\simDirichlet(1,…,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, \( Kn(\mathbf x)ek(\mathbf x)≤ n\choose k, \) 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 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/∏i xi, replace it by U/∏i 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 Width(In)\longrightarrow 2σz1-α/2\qquadalmost 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.