Aspects of Conditional Symmetry and Asymmetry of Copulas

Document Type : Research Paper

Authors

Department of Statistics, Yazd University

Abstract

‎‎The assumption of conditional symmetry‎, ‎which implies that the distribution of one random variable given another random variable has a symmetric form‎, ‎plays a crucial role in various probability and statistics problems‎. ‎This study examines copula properties related to the conditional symmetry/asymmetry of two random variables‎. ‎We investigate the possibility of ordering copulas based on their level of conditional asymmetry‎, ‎similar to the concordance ordering for dependence‎. ‎To quantify the degree of conditional asymmetry of a copula‎, ‎we introduce measures that are monotonic for the proposed ordering‎. ‎The characteristics of the proposed order and measures are elaborated upon‎. ‎Several examples are included to demonstrate the results‎. ‎

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References

[1] A. Alikhani-Vafa, A. Dolati, A measure of radial asymmetry for bivariate copulas based on Sobolev norm, Hacettepe
Journal of Mathematics and Statistics, 47(3) (2018), 649-658. https://doi.org/10.15672/HJMS.201613519886
[2] C. Amblard, S. Girard, Symmetry and dependence properties within a semiparametric family of bivariate copulas,
Nonparametric Statistics, 14(6) (2002), 715-727. https://doi.org/10.1080/10485250215322
[3] J. Bai, S. Ng, A consistent test for conditional symmetry in time series models, Journal of Econometrics, 103 (2001),
225-258. https://doi.org/10.1016/S0304-4076(01)00044-6
[4] S. Bouzebda, M. Cherfi, Test of symmetry based on copula function, Journal of Statistical Planning and Inference,
142(5) (2012), 1262-1271. https://doi.org/10.1016/j.jspi.2011.12.009
[5] T. Chen, G. Tripathi, A simple consistent test of conditional symmetry in symmetrically trimmed Tobit models,
Journal of Econometrics, 198 (2017), 29-40. https://doi.org/10.1016/j.jeconom.2016.12.003
[6] U. Cherubini, S. Mulinacci, F. Gobbi, S. Romagnoli, Dynamic copula methods in finance, John Wiley and Sons,
2011.
[7] A. Dehgani, A. Dolati, M. ´Ubeda-Flores, Measures of radial asymmetry for bivariate random vectors, Statistical
Papers, 54 (2013), 271-286. https://doi.org/10.1007/s00362-011-0425-y
[8] M. A. Delgado, X. Song, Nonparametric tests for conditional symmetry, Journal of Econometrics, 206(2) (2018),
447-471. https://doi.org/10.1016/j.jeconom.2018.06.010
[9] A. Dolati, A. Alikhani-Vafa, A copula-based index to measure directional reflection asymmetry for trivariate copulas,
Journal of The Iranian Statistical Society, 18(2) (2019), 139-153. https://doi.org/10.29252/jirss.18.2.139
[10] A. Dolati, M. Amini, G. R. Mohtashami Borzadaran, Some results on uncorrelated dependent random variables,
Journal of Mahani Mathematical Research, 11(3) (2022), 175-189. https://doi.org/10.22103/jmmr.2022.19416.
1245
[11] F. Durante, E. P. Klement, C. Sempi, M. ´Ubeda-Flores, Measures of non-exchangeability for bivariate random
vectors, Statistical Papers, 51 (2010), 687-699. https://doi.org/10.1007/s00362-008-0153-0
[12] M. Esfahani, M. Amini, G. R. Mohtashami-Borzadaran, A. Dolati, A new copula-based bivariate Gompertz–
Makeham model and its application to COVID-19 mortality data, Iranian Journal of Fuzzy Systems, 20(3) (2023),
159-175. https://doi.org/10.22111/ijfs.2023.7645
[13] Y. Gai, X. Zhu, J. Zhang, Testing symmetry of model errors for nonparametric regression models by using
correlation coefficient, Communications in Statistics-Simulation and Computation, 51(4) (2022), 1436-1453.
https://doi.org/10.1080/03610918.2019.1670844
[14] C. Genest, J. G. Ne˘slehov´a, On tests of radial symmetry for bivariate copulas, Statistical Papers, 55 (2014),
1107-1119. https://doi.org/10.1007/s00362-013-0556-4
[15] C. Genest, J. G. Ne˘slehov´a, L. F. Quessy, Tests of symmetry for bivariate copulas, Annals of the Institute of
Statistical Mathematics, 64(4) (2012), 811-834. https://doi.org/10.1007/s10463-011-0337-6
[16] H. Joe, Dependence modeling with copulas, CRC Press, 2014.
[17] P. Krupskii, Copula-based measures of refection and permutation asymmetry and statistical tests, Statistical Papers,
58 (2016), 1165-1187. https://doi.org/10.1007/s00362-016-0743-1
[18] E. Mokhtari, A. Dolati, A. Dastbaravarde, Copula-based measures and tests for conditional asymmetry, Communications in Statistics-Simulation and Computation, (2023), 1-22. https://doi.org/10.1080/03610918.2023.
2244703
[19] R. B. Nelsen, Some concepts of bivariate symmetry, Journal of Nonparametric Statistics, 3(1) (1993), 95-101.
https://doi.org/10.1080/10485259308832574
[20] R. B. Nelsen, An introduction to copulas, Second Edition, Springer, New York, 2006.
[21] R. B. Nelsen, Extremes of non-exchangeability, Statistical Papers, 48(2) (2007), 329-336. https://doi.org/10.
1007/s00362-006-0336-5
[22] J. F. Rosco, H. Joe, Measures of tail asymmetry for bivariate copulas, Statistical Papers, 54 (2013), 709-726.
https://doi.org/10.1007/s00362-012-0457-y
[23] M. Scarsini, On measures of concordance, Stochastica, 8 (1984), 201-218. http://eudml.org/doc/38916
[24] B. Schweizer, E. F. Wolff, On nonparametric measures of dependence for random variables, Annals of Statistics, 9
(1981), 879-885. https://doi.org/10.1214/aos/1176345528
[25] K. F. Siburg, K. Stehling, P. A. Stoimenov, G. N. F. Wei, An order of asymmetry in copulas, and implications
for risk management, Insurance: Mathematics and Economics, 68 (2016), 241-247. https://doi.org/10.1016/j.
insmatheco.2016.03.008
[26] A. Sklar, Functions de r´epartition `a n dimensions et leurs marges, Publications de L’In-stitut de Statistique de
L’Universite de Paris, 8 (1959), 229-231. https://hal.science/hal-04094463/document
[27] J. X. Zheng, Consistent specification testing for conditional symmetry, Econometric Theory, 14 (1998), 139-149.
https://doi.org/10.1017/S0266466698141063
Iranian Journal of Fuzzy Systems
Volume 21, Issue 6
November and December 2024
Pages 1-13

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