Assessing bivariate independence: Revisiting Bergsma's covariance

Bergsma (2006) proposed a covariance $\kappa$(X,Y) between random variables X and Y. He derived their asymptotic distributions under the null hypothesis of independence between X and Y . The non-null (dependent) case does not seem to have been studied in the literature. We derive several alternate expressions for $\kappa$. One of them leads us to a very intuitive estimator of $\kappa$(X,Y) that is a nice function of four naturally arising U-statistics. We derive the exact finite sample relation between all three estimates. The asymptotic distribution of our estimator, and hence also of the other two estimators, in the non-null (dependence) case, is then obtained by using the U-statistics central limit theorem. The exact value of $\kappa$(X,Y) is hard to calculate for most bivariate distributions. Here we provide detailed derivation of $\kappa$ for two well known parametric families, namely, the bivariate exponential and the bivariate normal distributions. Using these we carry out extensive simulation to study the properties of these estimates with a focus on the non-null case. In the null case, the limit is known to be degenerate. However with a higher scaling, the non-degenerate limit distribution of our estimator is again obtained using the theory of degenerate U-statistics. This quickly leads us also to the known asymptotic distribution results for the two estimates of Bergsma in the null case. We used simulation techniques for the null case to investigate the accuracies of the discrete approximation method.