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

By Marek Omelka, Irène Gijbels, Noël Veraverbeke. 2009

163 citations recorded by the source corpus.

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, Radulovic 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 comer 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, Radulovic 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.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/356ec4b465d70743a5a66acdcc22e249def80990

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Citations in the corpus

Estimation of a Conditional Copula and Association Measures (2011, methodology): “…estimation arise from the fact that the empirical distribution function estimator Hxh is evaluated at the estimated quantiles F −11xh(u1) and F −1 2xh(u2) leading to a double stochastic structure of the estimator Cxh. Similarly as in Omelka et al. (2009) we find the decomposition (A7) useful.”
Flexible Copula Density Estimation with Penalized Hierarchical B‐splines (2013, methodology): “As an alternative to wavelets, the use of Bernstein polynomials has been proposed in Sancetta and Satchell (2004); see also Qu, Qian, and Xie (2009) and Pfeifer, Straßburger, and Philipps (2009).”
Asymptotics of empirical copula processes under non-restrictive smoothness assumptions (2010, background): “For instance, the empirical copula is a basic building block when estimating copula densities (Chen and Huang [3], Omelka, Gijbels and Veraverbeke [22]) or dependence measures and functions (Schmid et al. [27], Genest and Segers [14]), for testing for independence (Genest and Rémillard [12], Genest,”
Copula-Based Regression Estimation and Inference (2013, background): “…for estimating c include kernel-smoothing estimators (Gijbels and Mielniczuk 1990; Charpentier, Fermanian, and Scaillet 2006; Chen and Huang 2007; Omelka, Gijbels, and Veraverbeke 2009), Bernstein estimator (Bouezmarni, Rom-bouts, and Taamouti 2010), and spline estimator (Kauermann et al. in…”
Probit Transformation for Nonparametric Kernel Estimation of the Copula Density (2014, methodology): “Moreover, Omelka et al (2009) explicitly show that they are satisﬁed by the Clayton, Gumbel, Gaussian and Student copulas.”
A Robust-Equitable Measure for Feature Ranking and Selection (2017, background): “This Hölder condition, while being a standard condition for density estimations, does not hold for any commonly used copula (Omelka et al., 2009; Segers, 2012).”
Estimation of a Copula when a Covariate Affects only Marginal Distributions (2015, result): “…that this result extends the standard results on weak convergence of the empirical copula estimator (see Fermanian et al., 2004; Tsukuhara, 2005; Omelka et al., 2009; Segers, 2012, among others) to the situation where residuals are used instead of independent and identically distributed…”
An Overview of the Goodness-of-Fit Test Problem for Copulas (2012, methodology): “One solution is to use improved kernel estimators that take care of the typical corner bias problem, as in [70].”
A Copula‐Based Non‐parametric Measure of Regression Dependence (2013, methodology): “In © 2012 Board of the Foundation of the Scandinavian Journal of Statistics. order to address the problem of boundary effects in the statistic ̂n, we have adapted the estimate investigated recently in Chen & Huang (2007) and Omelka et al. (2009) to our problem.”
A note on bootstrap approximations for the empirical copula process (2010, unknown): “(2004). See also Chen and Huang (2007) or Omelka, Gijbels and Veraverbeke (2009) for a smoothed”