The empirical discrete copula process

TL;DR

This paper develops a comprehensive inferential framework for discrete copulas on finite supports, establishing their properties and applications in statistical testing and dependence measurement.

Contribution

It introduces a novel approach defining discrete copulas via I-projection and derives their asymptotic properties, connecting to optimal transport theory.

Findings

Strong consistency and asymptotic normality of the empirical copula array.

Explicit covariance structure for the empirical copula estimator.

Application to large-sample distribution of Yule's coefficient and independence testing.

Abstract

This paper develops a general inferential framework for discrete copulas on finite supports in any dimension. The copula of a multivariate discrete distribution is defined as Csiszar's I-projection (i.e., the minimum-Kullback-Leibler divergence projection) of its joint probability array onto the polytope of uniform-margins probability arrays of the same size, and its empirical estimator is obtained by applying that same projection to the array of empirical frequencies observed on the sample. Under the assumption of random sampling, strong consistency and root-n-asymptotic normality of the empirical copula array is established, with an explicit "sandwich" form for its covariance. The theory is illustrated by deriving the large-sample distribution of Yule's concordance coefficient (the natural analogue of Spearman's rho for bivariate discrete distributions) and by constructing a test for quasi-independence in multivariate contingency tables. Our results not only complete the foundations of discrete-copula inference but also connect directly to entropically regularised optimal transport and other minimum-divergence problems.