## Electronic Journal of Statistics

### 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.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 2066-2096.

Dates
First available in Project Euclid: 19 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1495159235

Digital Object Identifier
doi:10.1214/17-EJS1280

Mathematical Reviews number (MathSciNet)
MR3652880

Zentralblatt MATH identifier
1395.62129

#### Citation

Bahraoui, Tarik; Quessy, Jean-François. Tests of radial symmetry for multivariate copulas based on the copula characteristic function. Electron. J. Statist. 11 (2017), no. 1, 2066--2096. doi:10.1214/17-EJS1280. https://projecteuclid.org/euclid.ejs/1495159235

#### References

• [1] Bàrdossy, A. (2006). Copula-based geostatistical models for groundwater quality parameters., Water Resources Research 42, 1–12.
• [2] Bouzebda, S. & Cherfi, M. (2012). Test of symmetry based on copula function., J. Statist. Plann. Inference 142, 1262–1271.
• [3] Dehling, H. & Mikosch, T. (1994). Random quadratic forms and the bootstrap for $U$-statistics., J. Multivariate Anal. 51, 392–413.
• [4] Demarta, S. & McNeil, A. J. (2005). The t copula and related copulas., International Statistical Review/Revue Internationale de Statistique, 111–129.
• [5] Dharmadhikari, S. & Joag-Dev, K. (1988)., Unimodality, convexity, and applications. Probability and Mathematical Statistics. Academic Press, Inc., Boston, MA.
• [6] Genest, C. & Nešlehová, J. G. (2014). On tests of radial symmetry for bivariate copulas., Statist. Papers 55, 1107–1119.
• [7] Henze, N., Klar, B. & Meintanis, S. G. (2003). Invariant tests for symmetry about an unspecified point based on the empirical characteristic function., J. Multivariate Anal. 87, 275–297.
• [8] Joe, H. (2015)., Dependence modeling with copulas, vol. 134 of Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL.
• [9] Kosorok, M. (2008)., Introduction to empirical processes and semiparametric inference. New York: Springer.
• [10] Lee, A. J. (1990)., $U$-statistics, vol. 110 of Statistics: Textbooks and Monographs. New York: Marcel Dekker Inc. Theory and practice.
• [11] Leucht, A. & Neumann, M. H. (2013). Dependent wild bootstrap for degenerate $U$- and $V$-statistics., J. Multivariate Anal. 117, 257–280.
• [12] Nelsen, R. B. (2006)., An introduction to copulas. Springer Series in Statistics. New York: Springer, 2nd ed.
• [13] Quessy, J.-F. (2016). A general framework for testing homogeneity hypotheses about copulas., Electron. J. Stat. 10, 1064–1097.
• [14] Quessy, J.-F., Rivest, L.-P. & Toupin, M.-H. (2016). On the family of multivariate chi-square copulas., J. Multivariate Anal..
• [15] Rosco, J. F. & Joe, H. (2013). Measures of tail asymmetry for bivariate copulas., Statist. Papers 54, 709–726.
• [16] Salvadori, G., de Michele, C., Kottegoda, N. T. & Rosso, R. (2007)., Extremes in nature: An approach using copulas. New York: Springer.
• [17] Segers, J. (2012). Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions., Bernoulli 18, 764–782.
• [18] Shorack, G. R. & Wellner, J. A. (1986)., Empirical processes with applications to statistics. New York: Wiley.
• [19] van der Vaart, A. W. & Wellner, J. A. (1996)., Weak convergence and empirical processes. Springer Series in Statistics. New York: Springer-Verlag. With applications to statistics.