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Multivariate copulas with hairpin support. (English) Zbl 1292.62086

Summary: The notion of a two-dimensional hairpin allows for two different extensions to the general multivariate setting–that of a sub-hairpin and that of a super-hairpin. We study existence and uniqueness of \(\rho\)-dimensional copulas whose support is contained in a sub- (or super-) hairpin and extend various results about doubly stochastic measures to the general multivariate setting. In particular, we show that each copula with hairpin support is necessarily an extreme point of the convex set of all \(\rho\)-dimensional copulas. Additionally, we calculate the corresponding Markov kernels and, using a simple analytic expression for sub- (or super-) hairpin copulas, analyze the strong interrelation with copulas having a fixed diagonal section. Several examples and graphics illustrate both the chosen approach and the main results.

MSC:

62H20 Measures of association (correlation, canonical correlation, etc.)
60E05 Probability distributions: general theory
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[1] Ahmad, N.; Kim, H. K.; McCann, R. J., Optimal transportation, topology and uniqueness, Bull. Math. Sci., 1, 1, 13-32 (2011) · Zbl 1255.49075
[2] Birkhoff, G., Lattice Theory, in: American Mathematical Society Colloquium Publications, vol. 25 (1948), American Mathematical Society: American Mathematical Society New York, N.Y
[3] Brown, J. R., Doubly stochastic measures and Markov operators, Michigan Math. J., 12, 367-375 (1965) · Zbl 0135.19103
[4] Cuculescu, I.; Theodorescu, R., Copulas: diagonals, tracks, Rev. Roumaine Math. Pures Appl., 46, 6, 731-742 (2001) · Zbl 1032.60009
[5] Durante, F.; Fernández-Sánchez, J., On the approximation of copulas via shuffles of Min, Statist. Probab. Lett., 82, 10, 1761-1767 (2012) · Zbl 1349.62174
[6] Durante, F.; Fernández-Sánchez, J.; Pappadá, R., Copulas, diagonals and tail dependence, Fuzzy Set Syst. (2014), (in press) http://dx.doi.org/10.1016/j.fss.2014.03.014 · Zbl 1360.68835
[7] Durante, F.; Fernández-Sánchez, J.; Sempi, C., Sklar’s theorem obtained via regularization techniques, Nonlinear Anal., 75, 2, 769-774 (2012) · Zbl 1229.62062
[8] Durante, F.; Sarkoci, P.; Sempi, C., Shuffles of copulas, J. Math. Anal. Appl., 352, 2, 914-921 (2009) · Zbl 1160.60307
[9] Fernández Sánchez, F.; Trutschnig, W., Some members of the class of (quasi-)copulas with given diagonal from the Markov kernel perspective, Comm. Statist. Theory Methods (2014), (in press)
[10] Fernández Sánchez, F.; Trutschnig, W., Conditioning based metrics on the space of multivariate copulas, their interrelation with uniform and levelwise convergence and Iterated Function Systems, J. Theoret Probab. (2014), (in press) http://dx.doi.org/10.1007/s10959-014-0541-4
[11] Fredricks, G. A.; Nelsen, R. B., Copulas constructed from diagonal sections, (Beneš, V.; Štěpán, J., Distributions with given marginals and moment problems (1996), Kluwer Acad. Publ., Dordrecht: Kluwer Acad. Publ., Dordrecht Prague), 129-136 · Zbl 0906.60022
[12] Hestir, K.; Williams, S. C., Supports of doubly stochastic measures, Bernoulli, 1, 3, 217-243 (1995) · Zbl 0844.60002
[13] Jaworski, P., On copulas and their diagonals, Inform. Sci., 179, 17, 2863-2871 (2009) · Zbl 1171.62332
[14] (Jaworski, P.; Durante, F.; Härdle, W. K., Copulae in Mathematical and Quantitative Finance. Copulae in Mathematical and Quantitative Finance, Lecture Notes in Statistics — Proceedings., vol. 213 (2013), Springer: Springer Berlin, Heidelberg) · Zbl 1268.91005
[15] (Jaworski, P.; Durante, F.; Härdle, W. K.; Rychlik, T., Copula Theory and its Applications. Copula Theory and its Applications, Lecture Notes in Statistics — Proceedings., vol. 198 (2010), Springer: Springer Berlin, Heidelberg) · Zbl 1194.62077
[16] Jaworski, P.; Rychlik, T., On distributions of order statistics for absolutely continuous copulas with applications to reliability, Kybernetika (Prague), 44, 6, 757-776 (2008) · Zbl 1180.60013
[17] Kallenberg, O., Foundations of Modern Probability (1997), Springer Verlag: Springer Verlag New York, Berlin, Heidelberg · Zbl 0892.60001
[18] Kamiński, A.; Mikusiński, P.; Sherwood, H.; Taylor, M. D., Doubly stochastic measures, topology, and latticework hairpins, J. Math. Anal. Appl., 152, 1, 252-268 (1990) · Zbl 0717.60007
[19] Kamiński, A.; Sherwood, H.; Taylor, M. D., Doubly stochastic measures with mass on the graphs of two functions, Real Anal. Exchange, 13, 1, 253-257 (1987/88) · Zbl 0639.60001
[20] Klenke, A., Probability Theory — A Comprehensive Course (2007), Springer Verlag: Springer Verlag Berlin, Heidelberg · Zbl 1141.60001
[21] Losert, V., Counterexamples to some conjectures about doubly stochastic measures, Pacific J. Math., 99, 2, 387-397 (1982) · Zbl 0468.28007
[22] Mai, J.-F.; Scherer, M., Lévy-frailty copulas, J. Multivariate Anal., 72, 3, 385-414 (2009) · Zbl 1162.62048
[23] Mai, J.-F.; Scherer, M., (Simulating Copulas. Simulating Copulas, Series in Quantitative Finance., vol.4 (2012), Imperial College Press: Imperial College Press London), Stochastic models, sampling algorithms, and applications
[25] Mai, J.-F.; Scherer, M., What makes dependence modeling challenging? Pitfalls and ways to circumvent them, Statist. Risk Modeling, 30, 4, 287-306 (2013) · Zbl 1287.91096
[26] Mikusiński, P.; Taylor, M. D., Some approximations of \(n\)-copulas, Metrika, 72, 3, 385-414 (2010) · Zbl 1197.62050
[27] Nelsen, R. B., An introduction to Copulas, (Springer Series in Statistics (2006), Springer: Springer New York) · Zbl 1152.62030
[28] Nelsen, R. B.; Fredricks, G. A., Diagonal copulas, (Beneš, V.; Štěpán, J., Distributions with Given Marginals and Moment Problems (1997), Kluwer Acad. Publ., Dordrecht), 121-128 · Zbl 0906.60021
[29] Peck, J. E.L., Doubly stochastic measures, Michigan Math. J., 6, 217-220 (1959) · Zbl 0090.26801
[30] Quesada Molina, J. J.; Rodríguez Lallena, J.-A., Some remarks on the existence of doubly stochastic measures with latticework hairpin support, Aequationes Math., 47, 2-3, 164-174 (1994) · Zbl 0798.60003
[31] Rudin, W., Real and Complex Analysis, (McGraw-Hill Series in Higher Mathematics (1987), McGraw-Hill Book Co: McGraw-Hill Book Co New York) · Zbl 0925.00005
[32] Rudin, W., Functional Analysis, (McGraw-Hill Series in Higher Mathematics (1991), McGraw-Hill Book Co: McGraw-Hill Book Co New York) · Zbl 0867.46001
[33] Rychlik, T., Sharp bounds on \(L\)-estimates and their expectations for dependent samples, Comm. Statist. Theory Methods, 22, 4, 1053-1068 (1993) · Zbl 0786.62046
[34] Rychlik, T., Distributions and expectations of order statistics for possibly dependent random variables, J. Multivariate Anal., 48, 1, 31-42 (1994) · Zbl 0790.62048
[35] Scarsini, M., Copulae of probability measures on product spaces, J. Multivariate Anal., 31, 201-219 (1989) · Zbl 0698.60016
[36] Seethoff, T. L.; Shiflett, R. C., Doubly stochastic measures with prescribed support, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 41, 4, 283-288 (1977/78) · Zbl 0364.60019
[37] Sherwood, H.; Taylor, M. D., Doubly stochastic measures with hairpin support, Probab. Theory Related Fields, 78, 4, 617-626 (1988) · Zbl 0629.60002
[38] Trutschnig, W., On a Strong Metric on the Space of Copulas and its Induced Dependence Measure, J. Math. Anal. Appl., 384, 690-705 (2011) · Zbl 1252.46019
[39] Trutschnig, W.; Fernández Sánchez, J., Some results on shuffles of two-dimensional copulas, J. Statist. Plann. Inference, 143, 2, 251-260 (2013) · Zbl 1268.62049
[40] Vitale, R. A., Approximation by mutually completely dependent processes, J. Approx. Theory, 66, 2, 225-228 (1991) · Zbl 0734.60038
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