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 marginal distribution of $Y$. They assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$ are independent, and it is $1$ exclusively for $Y$ being a 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 ${\mathrm{\Lambda }}_{\phi }$ induced by general convex functions $\phi :\mathbb{R}\to \mathbb{R}$, containing, e.g., Chatterjee’s coefficient of correlation as special case. After establishing additional useful properties of ${\mathrm{\Lambda }}_{\phi }$ we focus on continuous $(X,Y)$, translate ${\mathrm{\Lambda }}_{\phi }$ 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 construction underlying ${\mathrm{\Lambda }}_{\phi }$ can be used to define new measures of explainability generalizing the fraction of explained variance.