Gaussian limits for generalized spacings

TL;DR

This paper establishes Gaussian limit theorems for generalized spacings in multivariate samples, extending classical results and providing new asymptotic normality results for various divergence-based statistics.

Contribution

It introduces a unified framework for the asymptotic normality of divergence coefficients based on generalized spacings in multivariate data.

Findings

Asymptotic normality of divergence coefficients for large samples

Extension of classical CLT to multivariate spacings

Application to statistics like log-likelihood ratios and information gain

Abstract

Nearest neighbor cells in $R^d,d\in\mathbb{N}$, are used to define coefficients of divergence ($\phi$-divergences) between continuous multivariate samples. For large sample sizes, such distances are shown to be asymptotically normal with a variance depending on the underlying point density. In $d=1$, this extends classical central limit theory for sum functions of spacings. The general results yield central limit theorems for logarithmic $k$-spacings, information gain, log-likelihood ratios and the number of pairs of sample points within a fixed distance of each other.