Large-Sample Theory for the Bergsma-Dassios Sign Covariance

TL;DR

This paper develops large-sample distribution theory for the empirical Bergsma-Dassios sign covariance, a measure that detects independence between variables, providing asymptotic null distributions and simulation validation.

Contribution

It introduces the asymptotic distribution theory for the empirical sign covariance, extending understanding of its behavior under independence for large samples.

Findings

Asymptotic null distributions derived for the empirical sign covariance.

Simulations show the limiting distributions approximate finite-sample behavior.

The sign covariance effectively tests independence in large samples.

Abstract

The Bergsma-Dassios sign covariance is a recently proposed extension of Kendall's tau. In contrast to tau or also Spearman's rho, the new sign covariance $\tau^*$ vanishes if and only if the two considered random variables are independent. Specifically, this result has been shown for continuous as well as discrete variables. We develop large-sample distribution theory for the empirical version of $\tau^*$. In particular, we use theory for degenerate U-statistics to derive asymptotic null distributions under independence and demonstrate in simulations that the limiting distributions give useful approximations.