On the empirical multilinear copula process for count data

TL;DR

This paper studies the asymptotic properties of the empirical multilinear copula process for count data, providing convergence results and applications to independence testing in discrete multivariate settings.

Contribution

It establishes the asymptotic behavior of the empirical multilinear copula process for count data and introduces a consistent independence test applicable to high-dimensional sparse tables.

Findings

Process converges in $ ext{C}(K)$ for compact $K$ within a dense open subset of $[0,1]^d$

Results enable weak limit derivation for various functionals, including classical statistics

Develops a powerful independence test for sparse contingency tables

Abstract

Continuation refers to the operation by which the cumulative distribution function of a discontinuous random vector is made continuous through multilinear interpolation. The copula that results from the application of this technique to the classical empirical copula is either called the multilinear or the checkerboard copula. As shown by Genest and Ne\v{s}lehov\'{a} (Astin Bull. 37 (2007) 475-515) and Ne\v{s}lehov\'{a} (J. Multivariate Anal. 98 (2007) 544-567), this copula plays a central role in characterizing dependence concepts in discrete random vectors. In this paper, the authors establish the asymptotic behavior of the empirical process associated with the multilinear copula based on $d$-variate count data. This empirical process does not generally converge in law on the space $\mathcal {C}([0,1]^d)$ of continuous functions on $[0,1]^d$, equipped with the uniform norm. However, the authors show that the process converges in $\mathcal{C}(K)$ for any compact $K\subset\mathcal{O}$, where $\mathcal{O}$ is a dense open subset of $[0,1]^d$, whose complement is the Cartesian product of the ranges of the marginal distribution functions. This result is sufficient to deduce the weak limit of many functionals of the process, including classical statistics for monotone trend. It also leads to a powerful and consistent test of independence which is applicable even to sparse contingency tables whose dimension is sample size dependent.