A measure of radial asymmetry for bivariate copulas based on Sobolev norm

Year 2018, Volume: 47 Issue: 3, 649 - 658, 01.06.2018

Ahmad Alikhani-vafa Ali Dolati

https://izlik.org/JA46AB38HS

Abstract

The modified Sobolev norm is used to construct an index for measuring the degree of radial asymmetry of a copula. We study various aspects of this index and discuss its rank-based estimator. Through simulation and a real data example, we compare the proposed index with the other radial asymmetry measures.

Keywords

Bivariate symmetry , Copula , Empirical process , Radial asymmetry , Sobolev norm

References

• Bouzebda, S. and Cher, M. Test of symmetry based on copula function, Journal of Statistical Planning and Inference 142, 1262-1271, 2012.
• Darsow, W. F. and Olsen, E. T. Norms for copulas, International Journal of Mathematics and Mathematical Sciences 18, 417-436, 1995.
• Dehgani, A. Dolati, A., and Úbeda-Flores, M. Measures of radial asymmetry for bivariate random vectors, Statistical Papers 54, 271-286, 2013.
• Denuit, M., Purcaru, O. and Van Keilegom, I. Bivariate Archimedean copula modelling for loss-ALAE data in non-life insurance, Working paper, UCL, 2004.
• Durante, F. and Mesiar, R. L1-measure of non-exchangeability for bivariate extreme value and Archimax copulas, Journal of Mathematical Analysis and Applications 369, 610-615, 2010.
• Fermanian, J. D. and Scaillet, O. Nonparametric estimation of copulas for time series, Journal of Risk 5, 25-54, 2003.
• Fermanian, J. D., Radulovic, D. and Wegkamp, M. Weak convergence of empirical copula processes, Bernoulli 10 (5), 847860, 2004.
• Frees, E. and Valdez, E. Understanding relationships using copulas, North American Actuarial Journal 2, 1-25, 1998.
• Klugman, S. and Parsa, R. Fitting bivariate loss distributions with copulas, Insurance: Mathematics and Economics 24, 139-148, 1999.
• Genest, C. and Neslehova, J. G. On tests of radial symmetry for bivariatecopulas, Statistical Papers 55, 1107-1119, 2014.
• ojasiewicz, S. Introduction to the Theory of Real Functions, Wiley, Chichester, 1988.
• Nelsen, R.B. Some concepts of bivariate symmetry, Journal of Nonparametric Statistics 3, 95-101, 1993.
• Nelsen, R.B. An Introduction to Copulas. Second Edition, Springer, New York, 2006.
• Nelsen, R.B. Extremes of nonexchangeability, Statistical Papers 48, 329-336, 2007.
• Rosco, J. F. and Joe, H. Measures of tail asymmetry for bivariate copulas, Statistical Papers 4, 709-726, 2013.
• Segers, J. Asymptotics of empirical copula processes under non-restrictive smoothness as- sumptions, Bernoulli 18, 764-782, 2012.
• Siburg, K.F. and Stoimenov, P. A. A scalar product for copulas, Journal of Mathematical Analysis and Applications 344(1), 429-439, 2008.
• Siburg, K. F. and Stoimenov, P. A. Symmetry of functions and exchangeability of random variables, Statical Papers 52, 1-15, 2011.
• Sklar, A. Functions de répartition a n dimensions et leurs marges, Publ. Inst. Statistique Univ. Paris 8, 229-231, 1959.