Abstract
The unique copula of a continuous random pair \((X,Y)\) is said to be radially symmetric if and only if it is also the copula of the pair \((-X,-Y)\). This paper revisits the recently considered issue of testing for radial symmetry. Three rank-based statistics are proposed to this end which are asymptotically equivalent but simpler to compute than those of Bouzebda and Cherfi (J Stat Plan Inference 142:1262–1271, 2012). Their limiting null distribution and its approximation using the multiplier bootstrap are discussed. The finite-sample properties of the resulting tests are assessed via simulations. The asymptotic distribution of one of the test statistics is also computed under an arbitrary alternative, thereby correcting an error in the recent work of Dehgani et al. (Stat Pap 54:271–286, 2013).
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References
Bouzebda S, Cherfi M (2012) Test of symmetry based on copula function. J Stat Plan Inference 142:1262–1271
Bücher A, Dette H (2010) A note on bootstrap approximations for the empirical copula process. Stat Probab Lett 80:1925–1932
Dehgani A, Dolati A, Úbeda-Flores M (2013) Measures of radial asymmetry for bivariate random vectors. Stat Pap 54:271–286
Fermanian JD, Radulović D, Wegkamp M (2004) Weak convergence of empirical copula processes. Bernoulli 10:847–860
Gänßler P, Stute W (1987) Seminar on empirical processes. Birkhäuser, Basel
Genest C, Nešlehová JG (2013) Assessing and modeling asymmetry in bivariate continuous data. In: Jaworski P, Durante F, Härdle WK (eds) Copulae in mathematical and quantitative finance, Proceedings of the workshop held in Cracow, 10–11 July 2012. Springer, Berlin, pp 91–114
Genest C, Huang W, Dufour JM (2013) A regularized goodness-of-fit test for copulas. J SFdS 154:64–77
Genest C, Nešlehová J, Quessy JF (2012) Tests of symmetry for bivariate copulas. Ann Inst Stat Math 64:811–834
McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management: concepts, techniques and tools. Princeton University Press, Princeton
Nelsen RB (2006) An introduction to copulas, 2nd edn. Springer, New York
Neuhaus G, Zhu LX (1998) Permutation tests for reflected symmetry. J Multivar Anal 67:129–153
Ngatchou-Wandji J (2009) Testing for symmetry in multivariate distributions. Stat Methodol 6:230–250
Raghavachari M (1973) Limiting distributions of Kolmogorov–Smirnov type statistics under the alternative. Ann Stat 1:67–73
Rémillard B, Scaillet O (2009) Testing for equality between two copulas. J Multivar Anal 100:377–386
Rosco JF, Joe H (2013) Measures of tail asymmetry for bivariate copulas. Stat Pap 54:709–726
Rüschendorf L (1976) Asymptotic distributions of multivariate rank order statistics. Ann Stat 4:912–923
Segers J (2012) Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18:764–782
Sklar A (1959) Fonctions de répartition à \(n\) dimensions et leurs marges. Publ Inst Stat Univ Paris 8:229–231
van der Vaart AW, Wellner JA (1996) Weak convergence and empirical processes, with applications to statistics. Springer, New York
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This research was supported by the Canada Research Chairs Program and grants from the Natural Sciences and Engineering Research Council of Canada and the Fonds de recherche du Québec—Nature et technologies.
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Genest, C., Nešlehová, J.G. On tests of radial symmetry for bivariate copulas. Stat Papers 55, 1107–1119 (2014). https://doi.org/10.1007/s00362-013-0556-4
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DOI: https://doi.org/10.1007/s00362-013-0556-4