Assessing bivariate independence: Revisiting Bergsma's covariance

Abstract

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. For specific parametric bivariate distributions, the value of $\kappa$ can be derived in terms of the natural dependence parameters of these distributions. In particular, we derive the formula for $\kappa$ when (X,Y) are distributed as Gumbel's bivariate exponential. We bring out various aspects of these estimators through extensive simulations from several prominent bivariate distributions. In particular, we investigate the empirical relationship between $\kappa$ and the dependence parameters, the distributional properties of the estimators, and the accuracy of these estimators. We also investigate the powers of these measures for testing independence, compare these among themselves, and with other well known such measures. Based on these exercises, the proposed estimator seems as good or better than its competitors both in terms of power and computing efficiency.