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.

翻译：近期建立的针对随机变量对$(X,Y)$的定向依赖度量，基于比较给定$X=x$时$Y$的条件分布与$Y$的边缘分布这一自然思想。这些度量赋予$(X,Y)$对$[0,1]$区间内的数值：当且仅当$X,Y$独立时取值为$0$，而仅当$Y$为$X$的函数时取值为$1$。本文证明，若将随机抽样的条件分布相互比较，或等价地分析$Y$在给定$X=x$时条件分布对$x$的敏感程度，即可为构建由一般凸函数$\varphi: \mathbb{R} \rightarrow \mathbb{R}$诱导的新型依赖性度量族$\Lambda_\varphi$开辟新路径，该族包含Chatterjee相关系数作为特例。在建立$\Lambda_\varphi$的附加有用性质后，我们聚焦于连续型$(X,Y)$，将$\Lambda_\varphi$转化为copula框架，考虑其$L^p$版本，并建立具有完全一般性强相合性的估计量。通过真实数据实例和模拟研究验证了所选方法及估计量的性能。作为上述结果的补充，我们展示了如何通过轻微修改$\Lambda_\varphi$的底层构造来定义新的可解释性度量，推广解释方差的比例。