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.
Highlights
A random vector X = (X1, . . . , Xd) is said to be symmetric about a point μ = (μ1, . . . , μd) ∈ Rd if X − μ and μ − X have the same distribution
Of interest in this work is the relationship that exists between this notion of multivariate symmetry and the copula that can be extracted from the distribution of a random vector
The radial symmetry of C means that U ∼ C and 1d − U have the same distribution, where 1d = (1, . . . , 1) ∈ Rd
Summary
It is a good idea to test the radial symmetry hypothesis before trying to fit a specific copula model to multivariate observations. It is only in the case of its non rejection that the use of a family of radially symmetric copulas would be justified, e.g. the well-known Normal and Student copulas, or more generally the models in the elliptical class. Tests of radial symmetry based on empirical copulas have recently been proposed by [2] and [6] in the special case when d = 2; see [15] for the definition of measures of bivariate radial asymmetry These authors adopt a distribution-oriented perspective based on the comparison of nonparametric estimators of the copulas of (U1, U2) and (1 − U1, 1 − U2), respectively. All the proofs are relegated to Appendix A and some complementary computations are to be found in Appendix B
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