Mutual association measures

Abstract

Given two continuous variables X and Y with copula C,  many attempts were made in the literature to provide a suitable representation of the set of all concordance measures of the couple (X, Y), as defined by some axioms. Some papers concentrated on the need to let such measures vanish not only at independence but, more generally, at indifference, i.e. in lack of any dominance of concordance or discordance in (X, Y). The concept of indifference (or reflection-invariance in the language of copulas) led eventually to full characterizations of some well-defined subsets of measures. However, the built classes failed to contain some known measures, which cannot be regarded as averaged “distances” of C from given reference copulas (markedly, the Frechét–Hoeffding bounds). This paper, then, proposes a method to enlarge the representation of concordance measures, so as to include such elements, denoted here as mutual, which arise from averaged multiple comparisons of the copula C itself. Mutual measures, like Kendall’s tau and a recent proposal based on mutual variability, depend on the choice of a generator function, which needs some assumptions listed here. After defining some natural estimators for the elements of the enlarged class, the assumptions made on the generator are shown to guarantee that those statistics possess desirable sample properties, such as asymptotic normality under independence. Distributions under contiguous alternatives and relative efficiencies, in addition, are derived under mild assumptions on the copula. The paper provides several examples, giving further insight to some comparisons formerly conducted in the literature only via simulation.