Asymptotic Distribution for Symmetric Spacing Statistics

Abstract

A wide class of statistical problems is related to finding the asymptotic distribution of the following symmetric spacing statistic

where G is a continuous distribution function and X _1,n  ≤ ··· ≤ X _n,n are the order statistics of n independent observations from some continuous distribution F. When G = F, we provide a simple proof of a conjecture of Darling (1953) about asymptotic normality of T _φ,n (G) under sharp conditions on φ. When G ≠ F, we also obtain central limit theorems with explicit formulas of the asymptotic mean and variance for general φ. These results supersede the existing results and are discussed inside a concise review of the topic.

Keywords:

• φ-divergences
• Asymptotic normality
• Goodness-of-fit
• Spacings

Mathematics Subject Classification:

• Primary 62B10
• Secondary 62E20

Notes

Where γ = 0.57722… is the Euler's constant.