Large-Sample Theory for the Bergsma-Dassios Sign Covariance

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.