Constructing Measures of Dependence Via Sensitivity of Conditional Distributions

An essential objective in statistics is to determine the relationship, specifically the dependence, between two random variables X and Y. To this end many classic and novel methods have been described. Here we show that some of these methods can be seen as special cases of a broad class of measures of dependence, which essentially measure the sensitivity of the distribution of Y conditional on X to changes in X. We introduce some new members of this class which characterise dependence in the sense that they are equal to 0 if and only if X and Y are independent and are equal to 1 if and only Y is a measurable function of X. We establish some further useful properties of these measures and discuss their estimation. In particular, for continuous (X, Y), the introduced measures of dependence rely only on the underlying copula of (X, Y) and not on the marginal distributions. This allows the use of Checkerboard Copulas to obtain a strongly consistent estimator.