On grade transformation and its implications for copulas

As multivariate distributions with uniform one-dimensional margins, copulas provide very convenient models for studying dependence structure with tools that are scale-free. Each copula (n-copula) represents the whole class of continuous bivariate (multivariate) distributions from which it has been obtained when one-dimensional marginals were transformed by their cdf’s. The similar property, however, does not hold when the original distributions are discrete, or mixed discrete-continuous. After the transformation by marginal cdf’s the copula is not uniquely defined and consequently cannot be used for dependence studies analogously as in the continuous case. In this paper we will discuss grade transformation which, being an extension of probability integral transform, enables unique copula (or n-copula) representation of a multivariate distribution of any type for which cdf can be defined. We will also present how formulas of Kendall’s τ , Spearman’s ρ, and Gini correlation can be written for variables continuous or not using smoothed cdf’s instead of original ones.