Estimating scale-invariant directed dependence of bivariate distributions

Abstract

Asymmetry of dependence is an inherent property of bivariate probability distributions. Being symmetric, commonly used dependence measures such as Pearson’s r or Spearman’s ρ mask asymmetry and implicitly assume that a random variable Y is equally dependent on a random variable X as vice versa. A copula-based, hence scale-invariant dependence measure called ζ1 overcoming the just mentioned problem was introduced in 2011. ζ1 attains values in [0,1], it is 0 if, and only if X and Y are independent, and 1 if, and only if Y is a measurable function of X. Working with so-called empirical checkerboard copulas allows to construct an estimator ζ1n for ζ1 which is strongly consistent in full generality, i.e., without any smoothness assumptions on the underlying copula. The R-package qad (short for quantification of asymmetric dependence) containing the estimator ζ1n is used both, to perform a simulation study illustrating the small sample performance of the estimator as well as to estimate the directed dependence between some global climate variables as well as between world development indicators.