Quantifying and estimating dependence via sensitivity of conditional distributions

TL;DR

This paper introduces a new family of dependence measures based on the sensitivity of conditional distributions, generalizing existing metrics and providing consistent estimators with practical applications demonstrated through data analysis and simulations.

Contribution

It proposes novel dependence measures $\Lambda_\varphi$ derived from analyzing the sensitivity of conditional distributions, extending previous measures like Chatterjee's coefficient, and develops consistent estimators for continuous variables.

Findings

New dependence measures $\Lambda_\varphi$ generalize existing metrics.

Established strongly consistent estimators for these measures.

Illustrated effectiveness through real data and simulation studies.

Abstract

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 $\Lambda_\varphi$ induced by general convex functions $\varphi: \mathbb{R} \rightarrow \mathbb{R}$, containing, e.g., Chatterjee's coefficient of correlation as special case. After establishing additional useful properties of $\Lambda_\varphi$ we focus on continuous $(X,Y)$, translate $\Lambda_\varphi$ 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 $\Lambda_\varphi$ can be used to define new measures of explainability generalizing the fraction of explained variance.