- Volume 23, Number 3 (2017), 1848-1873.
Probit transformation for nonparametric kernel estimation of the copula density
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.
Bernoulli, Volume 23, Number 3 (2017), 1848-1873.
Received: October 2014
Revised: August 2015
First available in Project Euclid: 17 March 2017
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Geenens, Gery; Charpentier, Arthur; Paindaveine, Davy. Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli 23 (2017), no. 3, 1848--1873. doi:10.3150/15-BEJ798. https://projecteuclid.org/euclid.bj/1489737627
- Supplement to “Probit transformation for nonparametric kernel estimation of the copula density”. An appendix consisting of the proofs of the different theoretical results is available as supplementary material.