Adaptive Series Estimators for Copula Densities

In this thesis, based on an orthonormal series expansion, we propose a new nonparametric method to estimate copula density functions. Since the basis coefficients turn out to be expectations, empirical averages are used to estimate these coefficients. We propose estimators of the variance of the estimated basis coefficients and establish their consistency. We derive the asymptotic distribution of the estimated coefficients under mild conditions. We derive a simple oracle inequality for the copula density estimator based on a finite series using the estimated coefficients. We propose a stopping rule for selecting the number of coefficients used in the series and we prove that this rule minimizes the mean integrated squared error. In addition, we consider hard and soft thresholding techniques for sparse representations. We obtain oracle inequalities that hold with prescribed probability for various norms of the difference between the copula density and our threshold series density estimator. Uniform confidence bands are derived as well. The oracle inequalities clearly reveal that our estimator adapts to the unknown degree of sparsity of the series representation of the copula density. A simulation study indicates that our method is extremely easy to implement and works very well, and it compares favorably to the popular kernel based copula density estimator, especially around the boundary points, in terms of mean squared error. Finally, we have applied our method to an insurance dataset. After comparing our method with the previous data analyses, we reach the same conclusion as the parametric methods in the literature and as such we provide additional justification for the use of the developed parametric model.