Open Access
August 2017 Probit transformation for nonparametric kernel estimation of the copula density
Gery Geenens, Arthur Charpentier, Davy Paindaveine
Bernoulli 23(3): 1848-1873 (August 2017). DOI: 10.3150/15-BEJ798


Copula modeling has become ubiquitous in modern statistics. Here, the problem of nonparametrically estimating a copula density is addressed. Arguably the most popular nonparametric density estimator, the kernel estimator is not suitable for the unit-square-supported copula densities, mainly because it is heavily affected by boundary bias issues. In addition, most common copulas admit unbounded densities, and kernel methods are not consistent in that case. In this paper, a kernel-type copula density estimator is proposed. It is based on the idea of transforming the uniform marginals of the copula density into normal distributions via the probit function, estimating the density in the transformed domain, which can be accomplished without boundary problems, and obtaining an estimate of the copula density through back-transformation. Although natural, a raw application of this procedure was, however, seen not to perform very well in the earlier literature. Here, it is shown that, if combined with local likelihood density estimation methods, the idea yields very good and easy to implement estimators, fixing boundary issues in a natural way and able to cope with unbounded copula densities. The asymptotic properties of the suggested estimators are derived, and a practical way of selecting the crucially important smoothing parameters is devised. Finally, extensive simulation studies and a real data analysis evidence their excellent performance compared to their main competitors.


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Gery Geenens. Arthur Charpentier. Davy Paindaveine. "Probit transformation for nonparametric kernel estimation of the copula density." Bernoulli 23 (3) 1848 - 1873, August 2017.


Received: 1 October 2014; Revised: 1 August 2015; Published: August 2017
First available in Project Euclid: 17 March 2017

zbMATH: 06714321
MathSciNet: MR3624880
Digital Object Identifier: 10.3150/15-BEJ798

Keywords: Boundary bias , copula density , local likelihood density estimation , transformation kernel density estimator , unbounded density

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 3 • August 2017
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