Uniform in bandwidth law of the iterated logarithm for a transformation kernel estimator of copulas

Abstract

In this paper, we establish a uniform in bandwidth law of the iterated logarithm for the Transformation kernel estimator of bivariate copulas introduced in Omelka et al. (2009). To this end, we make use of a general empirical process approach inspired by the works in Mason and Swanepoel (2011). We obtain the asymptotic order of the maximal deviation of this estimator from its expectation. Then, we show that the bias converges asymptotically to zero at the same order provided that the second-order partial derivatives of the copula exist and are bounded.We also propose a bandwidth selection method by using a cross-validation approach. Finally, we compare in a simulation study the performances of the Transformation kernel estimator by considering two different methods of selecting the bandwidth.

Keywords: Copula function; Kernel estimation; Transformation estimator; Law of the iterated logarithm; Uniform in bandwidth consistency; Cross-validation method