The Annals of Statistics

Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing

Marek Omelka, Irène Gijbels, and Noël Veraverbeke

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We reconsider the existing kernel estimators for a copula function, as proposed in Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445–464], Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847–860] and Chen and Huang [Canad. J. Statist. 35 (2007) 265–282]. All of these estimators have as a drawback that they can suffer from a corner bias problem. A way to deal with this is to impose rather stringent conditions on the copula, outruling as such many classical families of copulas. In this paper, we propose improved estimators that take care of the typical corner bias problem. For Gijbels and Mielniczuk [Comm. Statist. Theory Methods 19 (1990) 445–464] and Chen and Huang [Canad. J. Statist. 35 (2007) 265–282], the improvement involves shrinking the bandwidth with an appropriate functional factor; for Fermanian, Radulovič and Wegkamp [Bernoulli 10 (2004) 847–860], this is done by using a transformation. The theoretical contribution of the paper is a weak convergence result for the three improved estimators under conditions that are met for most copula families. We also discuss the choice of bandwidth parameters, theoretically and practically, and illustrate the finite-sample behaviour of the estimators in a simulation study. The improved estimators are applied to goodness-of-fit testing for copulas.

Article information

Ann. Statist. Volume 37, Number 5B (2009), 3023-3058.

First available in Project Euclid: 17 July 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Copula Cramér–von Mises statistics Gaussian process goodness-of-fit Kendall’s tau Kolmogorov–Smirnov statistics parametric bootstrap pseudo-observations weak convergence


Omelka, Marek; Gijbels, Irène; Veraverbeke, Noël. Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing. Ann. Statist. 37 (2009), no. 5B, 3023--3058. doi:10.1214/08-AOS666.

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