Quantifying and estimating dependence via sensitivity of conditional distributions

Recently established, directed dependence measures for pairs (X,Y) of random variables build upon the natural idea of comparing the conditional distributions of Y given X = x with the unconditional distribution of Y. They assign pairs (X,) values in [0, 1], where the value is 0 if and only if X, are independent, and it is 1 exclusively for Y being a measurable function of X. Here we show that comparing randomly drawn conditional distributions with each other instead or, equivalently, analyzing how sensitive the conditional distribution of Y given X = x is on x, opens the door to constructing novel families of dependence measures Λφ induced by general convex functions φ: R → R, containing, e.g., Chatterjee’s coefficient of correlation as special case. After establishing additional useful properties of Λφ we focus on continuous (X,Y), translate Λφ to the copula setting, consider the L ^p-version and establish an estimator which is strongly consistent in full generality. A real data example and a simulation study illustrate the chosen approach and the performance of the estimator. Complementing the afore-mentioned results, we show how a slight modification of the sensitivity idea underlying Λφ can be used to define new measures of explainability generalizing the fraction of explained variance.