Goodness‐of‐fit Procedures for Copula Models Based on the Probability Integral Transformation

Wang & Wells [ J. Amer. Statist. Assoc. 95 (2000) 62] describe a non‐parametric approach for checking whether the dependence structure of a random sample of censored bivariate data is appropriately modelled by a given family of Archimedean copulas. Their procedure is based on a truncated version of the Kendall process introduced by Genest & Rivest [ J. Amer. Statist. Assoc. 88 (1993) 1034] and later studied by Barbe et al . [ J. Multivariate Anal. 58 (1996) 197]. Although Wang & Wells (2000) determine the asymptotic behaviour of their truncated process, their model selection method is based exclusively on the observed value of its L 2 ‐norm. This paper shows how to compute asymptotic p ‐values for various goodness‐of‐fit test statistics based on a non‐truncated version of Kendall's process. Conditions for weak convergence are met in the most common copula models, whether Archimedean or not. The empirical behaviour of the proposed goodness‐of‐fit tests is studied by simulation, and power comparisons are made with a test proposed by Shih [ Biometrika 85 (1998) 189] for the gamma frailty family.