Smooth test for equality of copulas

A smooth test to simultaneously compare K copulas, where K ≥ 2 is proposed. The K observed populations can be paired, and the test statistic is constructed based on the differences between moment sequences, called copula coefficients. These coefficients characterize the copulas, even when the copula densities may not exist. The procedure employs a two-step data-driven procedure. In the initial step, the most significantly different coefficients are selected for all pairs of populations. The subsequent step utilizes these coefficients to identify populations that exhibit significant differences. To demonstrate the effectiveness of the method, we provide illustrations through numerical studies and application to two real datasets.