K-Sample Test for Equality of Copulas

We propose a test procedure to compare simultaneously $K$ copulas, with $K \geq 2$. The $K$ observed populations can be paired. The test statistic is based on the differences between orthogonal projection coefficients associated to the density copulas, that we called {\it copula coefficients}. The procedure is data driven and we obtain a chi-square asymptotic distribution of the test statistic under the null. We illustrate our procedure via numerical studies and through two real datasets. Eventually, a clustering algorithm is deduced from the $K$-sample test and its performances are illustrated in a simulation experiment.