Bernoulli

  • Bernoulli
  • Volume 20, Number 3 (2014), 1344-1371.

On the empirical multilinear copula process for count data

Christian Genest, Johanna G. Nešlehová, and Bruno Rémillard

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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 Nešlehová (Astin Bull. 37 (2007) 475–515) and Nešlehová (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.

Article information

Source
Bernoulli, Volume 20, Number 3 (2014), 1344-1371.

Dates
First available in Project Euclid: 11 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.bj/1402488942

Digital Object Identifier
doi:10.3150/13-BEJ524

Mathematical Reviews number (MathSciNet)
MR3217446

Zentralblatt MATH identifier
1365.62221

Keywords
checkerboard copula contingency table count data empirical process Kendall’s tau mid-ranks multilinear extension copula Spearman’s rho test of independence

Citation

Genest, Christian; Nešlehová, Johanna G.; Rémillard, Bruno. On the empirical multilinear copula process for count data. Bernoulli 20 (2014), no. 3, 1344--1371. doi:10.3150/13-BEJ524. https://projecteuclid.org/euclid.bj/1402488942


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