Multivariate radial symmetry of copula functions: finite sample comparison in the i.i.d case. (English) Zbl 1474.62195
Summary: Given a \(d\)-dimensional random vector \(\textbf{X} = (X_1, \dots, X_d)\), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point reflection through the center of the unit hypercube, then X is said to have copula radial symmetry. We generalize to higher dimensions the bivariate test introduced in [C. Genest and J. G. Nešlehová, Stat. Pap. 55, No. 4, 1107–1119 (2014; Zbl 1310.62069)], using three different possibilities for estimating copula derivatives under the null. In a comprehensive simulation study, we assess the finite-sample properties of the resulting tests, comparing them with the finite-sample performance of the multivariate competitors introduced in [P. Krupskii, Stat. Pap. 58, No. 4, 1165–1187 (2017; Zbl 1383.62162)] and [T. Bahraoui and J.-F. Quessy, Electron. J. Stat. 11, No. 1, 2066–2096 (2017; Zbl 1395.62129)].
MSC:
| 62H15 | Hypothesis testing in multivariate analysis |
| 62G10 | Nonparametric hypothesis testing |
| 62G30 | Order statistics; empirical distribution functions |
| 62H05 | Characterization and structure theory for multivariate probability distributions; copulas |
References:
| [1] | Bahraoui, T. and J.-F., Quessy (2017). Tests of radial symmetry for multivariate copulas based on the copula characteristic function. Electron. J. Stat. 11(1), 2066-2096. · Zbl 1395.62129 |
| [2] | Bouzebda, S. and M., Cherfi (2012). Test of symmetry based on copula function. J. Statist. Plann. Inference. 142(5), 1262-1271. · Zbl 1236.62037 |
| [3] | Bücher, A. and I., Kojadinovic (2016). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. Bernoulli. 22(2), 927-968. · Zbl 1388.62123 |
| [4] | Bücher, A. and S., Volgushev (2013). Empirical and sequential empirical copula processes under serial dependence. J. Multivariate Anal. 119, 61-70. · Zbl 1277.62223 |
| [5] | Carabarin-Aguirre, A. and B. G., Ivanoff (2010). Estimation of a distribution under generalized censoring. J. Multivariate Anal. 101(6), 1501-1519. · Zbl 1186.62048 |
| [6] | Dehgani, A., Dolati, A. and M., Úbeda Flores (2013). Measures of radial asymmetry for bivariate random vectors. Statist. Papers. 54(2), 271-286. · Zbl 1364.62116 |
| [7] | Demarta, S. and A. J., McNeil (2005) The t copula and related copulas. Int. Stat. Rev. 73(1), 111-129. · Zbl 1104.62060 |
| [8] | Einmahl, J. H. J. (1987). Multivariate Empirical Processes. PhD thesis, Centrum voor Wiskunde en Informatica, Amsterdam. · Zbl 0619.60031 |
| [9] | Fang, K. T., Kotz, S. and K. W., Ng (1990). Symmetric Multivariate and Related Distributions. Chapman and Hall, London. · Zbl 0699.62048 |
| [10] | Favre, A.-C., Quessy, J.-F. and M.-H., Toupin (2018). The new family of Fisher copulas to model upper tail dependence and radial asymmetry: properties and application to high-dimensional rainfall data. Environmetrics 29(3), Article ID e2494, 17 pages. |
| [11] | Genest, C. and J. G., Nešlehová (2014). On tests of radial symmetry for bivariate copulas. Statist. Papers. 55(4), 1107-1119. · Zbl 1310.62069 |
| [12] | Hammersley, J. M. and K. W., Morton (1956). A new Monte Carlo technique: antithetic variates. Proc. Cambridge Philos. Soc. 52, 449-475. · Zbl 0071.35404 |
| [13] | Hildebrandt, T. H. (1963) Introduction to the Theory of Integration. Academic Press, New York. · Zbl 0112.28302 |
| [14] | Hofert, M. (2008). Sampling Archimedean copulas. Comput. Statist. Data Anal. 52(12), 5163-5174. · Zbl 1452.62070 |
| [15] | Hofert, M., I., Kojadinovic, M., Maechler and J., Yan (2020). copula: Multivariate Dependence with Copulas. R package version 1.0-1. Available on CRAN. |
| [16] | Joe, H. (2014). Dependence Modeling with Copulas. CRC Press, Boca Raton FL. · Zbl 1346.62001 |
| [17] | Krupskii, P. (2017). Copula-based measures of reflection and permutation asymmetry and statistical tests. Statist. Papers. 58(4), 1165-1187. · Zbl 1383.62162 |
| [18] | Lee, W. and J. Y., Ahn (2014). On the multidimensional extension of countermonotonicity and its applications. Insurance Math. Econom. 56, 68-79. · Zbl 1304.62086 |
| [19] | Li, B. and M. G., Genton (2013). Nonparametric identification of copula structures. J. Amer. Statist. Assoc. 108(502), 666-675. · Zbl 06195969 |
| [20] | Nelsen, R. B. (1993). Some concepts of bivariate symmetry. J. Nonparametr. Statist. 3(1), 95-101. · Zbl 1380.62227 |
| [21] | Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York. · Zbl 1152.62030 |
| [22] | Nelsen, R. B. and M., Úbeda Flores (2012). Directional dependence in multivariate distributions. Ann. Inst. Statist. Math. 64(3), 677-685. · Zbl 1237.62076 |
| [23] | Quessy, J.-F. (2016). A general framework for testing homogeneity hypotheses about copulas. Electron. J. Stat. 10(1), 1064-1097. · Zbl 1336.62142 |
| [24] | Quessy, J.-F. and M., Durocher (2019). The class of copulas arising from squared distributions: properties and inference. Econom. Stat. 12, 148-166. |
| [25] | Rémillard, B. and O., Scaillet (2009). Testing for equality between two copulas. J. Multivariate Anal. 100(3), 377-386. · Zbl 1157.62401 |
| [26] | Rosco, J. F. and H., Joe, H. (2013). Measures of tail asymmetry for bivariate copulas. Statist. Papers. 54(3), 709-726. · Zbl 1307.62157 |
| [27] | Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli. 18(3), 764-782. · Zbl 1243.62066 |
| [28] | Sklar, A. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229-231. · Zbl 0100.14202 |
| [29] | van der Vaart, A. W. and J. A., Wellner (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002 |
| [30] | Weibel, M., D., Luethi and W., Breymann (2020). ghyp: Generalized Hyperbolic Distribution and Its Special Cases. R package version 1.6.1. Available on CRAN. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
