Tests of radial symmetry for multivariate copulas based on the copula characteristic function

Abstract

A new class of rank statistics is proposed to assess that the copula of a multivariate population is radially symmetric. The proposed test statistics are weighted $L_{2}$ functional distances between a nonparametric estimator of the characteristic function that one can associate to a copula and its complex conjugate. It will be shown that these statistics behave asymptotically as degenerate V-statistics of order four and that the limit distributions have expressions in terms of weighted sums of independent chi-square random variables. A suitably adapted and asymptotically valid multiplier bootstrap procedure is proposed for the computation of $p$-values. One advantage of the proposed approach is that unlike methods based on the empirical copula, the partial derivatives of the copula need not be estimated. The good properties of the tests in finite samples are shown via simulations. In particular, the superiority of the proposed tests over competing ones based on the empirical copula investigated by [6] in the bivariate case is clearly demonstrated.