In a recent paper Fermanian (2005)  studied a goodness-of-fit test for the parametric form of a copula, which is based on an L2-distance between a parametric and a nonparametric estimate of the copula density. In the present paper we investigate the asymptotic properties of the proposed test statistic under fixed alternatives. We also study the impact of different estimates for the parameters of the finite-dimensional family of copulas specified by the null hypothesis and illustrate the performance of a parametric bootstrap procedure for the approximation of the critical values. © 2009 Elsevier Inc. All rights reserved.
Bücher, A., & Dette, H. (2010). Some comments on goodness-of-fit tests for the parametric form of the copula based on L2-distances. Journal of Multivariate Analysis, 101(3), 749–763. https://doi.org/10.1016/j.jmva.2009.09.014