On a multivariate copula-based dependence measure and its estimation

TL;DR

This paper introduces a new copula-based multivariate dependence measure that is scale-invariant, analyzes its properties, and proposes a strongly consistent estimator with demonstrated small sample performance.

Contribution

It defines a novel dependence measure based on linkages, analyzes its properties, and develops a consistent checkerboard estimator applicable without smoothness assumptions.

Findings

Dependence measure is zero if and only if variables are independent.

Maximal dependence occurs only when Y is a function of X.

The estimator is strongly consistent and performs well in small samples.

Abstract

Working with so-called linkages allows to define a copula-based, $[0,1]$-valued multivariate dependence measure $\zeta^1(\boldsymbol{X},Y)$ quantifying the scale-invariant extent of dependence of a random variable $Y$ on a $d$-dimensional random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ which exhibits various good and natural properties. In particular, $\zeta^1(\boldsymbol{X},Y)=0$ if and only if $\boldsymbol{X}$ and $Y$ are independent, $\zeta^1(\boldsymbol{X},Y)$ is maximal exclusively if $Y$ is a function of $\boldsymbol{X}$, and ignoring one or several coordinates of $\boldsymbol{X}$ can not increase the resulting dependence value. After introducing and analyzing the metric $D_1$ underlying the construction of the dependence measure and deriving examples showing how much information can be lost by only considering all pairwise dependence values $\zeta^1(X_1,Y),\ldots,\zeta^1(X_d,Y)$ we derive a so-called checkerboard estimator for $\zeta^1(\boldsymbol{X},Y)$ and show that it is strongly consistent in full generality, i.e., without any smoothness restrictions on the underlying copula. Some simulations illustrating the small sample performance of the estimator complement the established theoretical results.