Nonparametric estimation of copula functions for dependence modelling

Copulas are full measures of dependence among components of random vectors. Unlike the marginal and the joint distributions, which are directly observable, a copula is a hidden dependence structure that couples a joint distribution with its marginals. This makes the task of proposing a parametric copula model non-trivial and is where a nonparametric estimator can play a significant role. In this paper, we propose a kernel estimator which is mean square consistent everywhere in the support of the copula function. The bias and variance of the copula estimator are derived which reveal the effects of kernel smoothing on the copula estimation. A smoothing bandwidth selection rule based on the derived bias and variance is proposed. The theoretical findings are confirmed by a simulation study. The kernel estimator is then used to formulate a goodness-of-fit test for parametric copula models. Title in French: we can supply this Résumé : Insérer votre résumé ici. Copulas are full measures of dependence among components of random vectors. Unlike the marginal and the joint distributions which are directly observable, a copula is a hidden dependence structure that couples a joint distribution with its marginals. This makes the task of proposing a parametric copula model non-trivial and is where a nonparametric estimator can play a significant role. In this paper, we propose a kernel estimator which is mean square consistent everywhere in the support of the copula function. The bias and variance of the copula estimator are derived which reveal the effects of kernel smoothing on the copula estimation. A smoothing bandwidth selection rule based on the derived bias and variance is proposed. The theoretical findings are confirmed by a simulation study. The kernel estimator is then used to formulate a goodness-of-fit test for parametric copula models.