Efficient Computation of the Bergsma-Dassios Sign Covariance

TL;DR

This paper presents an efficient algorithm for computing the Bergsma-Dassios sign covariance, reducing the computational complexity from quartic to near-quadratic time, enabling practical application to larger datasets.

Contribution

The authors develop a novel $O(n^2 \, \log n)$ algorithm for calculating the empirical Bergsma-Dassios sign covariance, significantly improving computational efficiency over previous methods.

Findings

Algorithm reduces computation from $O(n^4)$ to $O(n^2 \log n)$

Enables practical application of the sign covariance to large datasets

Provides a faster method for independence testing using $\tau^*$

Abstract

In an extension of Kendall's $\tau$, Bergsma and Dassios (2014) introduced a covariance measure $\tau^*$ for two ordinal random variables that vanishes if and only if the two variables are independent. For a sample of size $n$, a direct computation of $t^*$, the empirical version of $\tau^*$, requires $O(n^4)$ operations. We derive an algorithm that computes the statistic using only $O(n^2\log(n))$ operations.