Goodness-of-fit tests for the family of multivariate chi-square copulas

Nonparametric moment-based goodness-of-fit tests are developed for the family of chi-square copulas of arbitrary dimensions. This class of dependence models allows for tail asymmetries and contains the family of multivariate normal copulas as a special case. The proposed tests are based on two rank correlation coefficients whose population versions are equal, up to a monotone transformation, when the underlying dependence structure is a chi-square copula. The test statistics are computed from natural rank-based estimations of these two correlation coefficients and their large-sample distributions under the null hypothesis of a chi-square copula are derived; the validity of a parametric bootstrap procedure for the computation of p-values is formally established as well. Particular attention is given to tests for the families of normal and centered chi-square copulas. The simulations that are reported indicate that the new tests are reliable alternatives to those based on the empirical copula, both in the bivariate and multivariate cases. The usefulness of the introduced methodology is illustrated on the five-dimensional Nutrient dataset.