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 well-known Kendall’s τ correlation coeffcient,which is Bergsma-Dassios sign covariance.In this paper,a method for calculating the exact distribution of t^*,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^* when n=4,5,6,7,the exact variance of the projection of the kernel function of t^* can be calculated without solving the algebraic representation of the projection of the kernel function of t^*.Meanwhile,we utilized t^* and its exact variance to further investigate the hypothesis testing problem of examining the independence between two random variables.Finally,the simulation result verifies the accuracy of the exact distribution of t^* when n=4,5,6,7 and the validity of the hypothesis test.