How to Measure Dependence?

Abstract

The notion of dependence between two (or more) random variables is not a simple mathematical concept. Consequently, it is quite challenging to communicate information like the ‘degree’, ‘level’, or ‘type’ of dependence. A significant simplification is achieved if the information about the dependence structure is compressed into a single number that quantifies the degree of dependence on some scale ranging from —1 to + 1, say. Obviously, this comes at the price of losing information, since it corresponds to a mapping from the set of copulas to the real numbers. Moreover, there are several methods by which this aggregation of information is possible. It is very important to understand the philosophy behind those different methods, as each dependence measure quantifies only a certain aspect of the dependence structure. Obviously, our aim is to measure the respective aspect of dependence that is relevant for the application we have in mind. Taking the correlation between X₁ and X₂ as an example, this dependence measure quantifies the degree of linear dependence, see below. Apart from measuring various aspects of dependence, dependence measures can also be used to estimate the parameters of a parametric family of copulas. This application compares a theoretical dependence measure with the empirical counterpart and selects the parameters of the copula such that both (theoretical and empirical) measures agree. For several dependence measures we have empirical estimates with known finite sample (or asymptotic) distribution. This allows us, for example, to perform hypothesis tests.