Towards a universal representation of statistical dependence

Abstract

Dependence is undoubtedly a central concept in statistics. Though, it proves difficult to locate in the literature a formal definition which goes beyond the self-evident 'dependence = non-independence'. This absence has allowed the term 'dependence' and its declination to be used vaguely and indiscriminately for qualifying a variety of disparate notions, leading to numerous incongruities. For example, the classical Pearson's, Spearman's or Kendall's correlations are widely regarded as 'dependence measures' of major interest, in spite of returning 0 in some cases of deterministic relationships between variables of interest, evidently not measuring dependence at all. Arguing that research on such a fundamental topic would benefit from a slightly more rigid framework, this paper suggests a general definition of the dependence between two random variables defined on the same probability space. Natural enough for aligning with intuition, that definition is still sufficiently precise for allowing unequivocal identification of a 'universal' representation of the dependence structure of any bivariate distribution. Links between this representation and familiar concepts are highlighted, and ultimately, the idea of a dependence measure based on that universal representation is explored and shown to satisfy Rényi's postulates.

(Re-)Reading Sklar (1959)—A Personal View on Sklar’s Theorem

“…If the dependence parameter is to be 'margin-free', it must be (any one-to-one function of) the odds-ratio ω = p 00 p 11 p 10 p 01 ([24], Theorem 6.3 in [25]). It is thus fair to identify the dependence structure to the value of ω in this case-see Section 4.3.1 of Geenens [26] for a thorough discussion.…”