A New Algorithm for the Exact Null Variance of the Sign Covariance of Bergsma-Dassios

To measure the independence of two random variables, τ^* is proposed based on the well-known Kendall's τ correlation coefficient, which is Bergsma-Dassios sign covariance. In this paper, a method for calculating the exact distribution of t^\ast, the empirical version of τ^*, is given by using the red-black tree algorithm in the self-balancing binary search tree. Furthermore, by using the exact distribution of t^\ast when n=4, 5, 6, 7, the exact variance of the projection of the kernel function of t^\ast can be calculated without solving the algebraic representation of the projection of the kernel function of t^\ast. Meanwhile, we employ t^\ast and its exact variance to further investigate the hypothesis testing problem of examining the independence between two random variables. Finally, simulation results verify the accuracy of the exact distribution of t^\ast when n = 4, 5, 6, 7 and the validity of the hypothesis test.