A general framework for testing homogeneity hypotheses about copulas

Abstract

The dependence structure in a $d$-variate continuous random vector $\mathbf{X}$ is characterized by its unique copula. Starting from the fact that many copulas can be extracted from the global $d$-dimensional copula of $\mathbf{X}$, a very general framework is proposed here for testing that a given collection of induced $p$-dimensional copulas from a multivariate distribution are identical. Many hypotheses of interest in copula modeling fall into this category, including bivariate symmetry (diagonal, radial, joint), exchangeability, as well as various types of equality of copulas. Here, a broad class of test statistics is defined around a matrix representation of the null hypothesis and quadratic functionals including Cramér–von Mises and characteristic function mappings. Since the null hypotheses to be tested are composite by nature, the computation of $\mathrm{P}$-values is achieved using multiplier bootstrap versions of the test statistics. The sample properties of the method are investigated when testing for several types of bivariate symmetry, exchangeability, equality of non-overlapping and overlapping copulas and equality of all bivariate copulas. The general conclusion is that the tests are good at keeping their nominal level and are powerful against a wide variety of alternatives, showing the relevance and reliability of the methodology for the modeling of multivariate datasets with the help of copulas.