Weak convergence of empirical and bootstrapped C-power processes and application to copula goodness-of-fit

The selection of a copula that appropriately describes the dependence structure of a bivariate distribution function is an issue of a prime interest in multivariate statistical analysis. The most popular methods of testing are those based on the empirical copula and on Kendall’s process; these tests require the use of the parametric bootstrap, which is computationally heavy. This paper introduces the concept of C-power functions associated to bivariate copulas. In the first step, their theoretical properties are explored and computations for many popular dependence models are provided. Then, empirical counterparts of these functions are proposed and the weak convergence of suitably normalized versions is obtained from an application of the functional delta method. In addition, a multiplier bootstrap method for these C-power processes is described and validated. While of an independent interest to treat many hypotheses related to copulas, these newly introduced tools are used to develop a goodness-of-fit test for bivariate multi-parameter copula models. The recursive nature of the C-power functions induces easily computable formulas for the test statistics and suggests a sequential method of testing. The sample properties of this goodness-of-fit test are investigated and compared to those of competing procedures, and its usefulness is illustrated on some pairs of the multivariate Uranium exploration data set.