Estimating copula densities, using model selection techniques

Recently a new way of modeling dependence has been introduced considering a sequence of parametric copula models, covering more and more dependency aspects and approximating in this way the true copula density more and more. The method uses contamination families based on Legendre polynomials. It has been shown that in general after a few steps accurate approximations are obtained. In this paper selection of the adequate number of steps is treated, and estimation of the unknown parameters within the chosen contamination family is established. There should be a balance between the complexity of the model and the number of parameters to be estimated. High complexity gives a low model error, but a large stochastic or estimation error, while a very simple model gives a small stochastic error, but a large model error. Techniques from model selection are applied, thus letting the data tell us which aspects are important enough to capture into the model. Natural and simple estimators complete the procedure. Theoretical results show that the expected quadratic error is reduced by the selection rule to the same order of magnitude as in a classical parametric problem. The method is applied on a real data set, illustrating that the new method describes the data set very well: the error involved by the classical Gaussian copula is reduced with no fewer than 50%.