Equitability of Dependence Measure

Measuring dependence between two random variables is very important, and critical in many applied areas such as variable selection, brain network analysis. However, we do not know what kind of functional relationship is between two covariates, which requires the dependence measure to be equitable. That is, it gives similar scores to equally noisy relationship of different types. In fact, the dependence score is a continuous random variable taking values in $[0,1]$, thus it is theoretically impossible to give similar scores. In this paper, we introduce a new definition of equitability of a dependence measure, i.e, power-equitable (weak-equitable) and show by simulation that HHG and Copula Dependence Coefficient (CDC) are weak-equitable.