Quantifying and estimating dependence via sensitivity of conditional distributions
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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 Λ_φ 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 Λ_φ we focus on continuous (X,Y), translate Λ_φ 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 Λ_φ can be used to define new measures of explainability generalizing the fraction of explained variance.
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