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Statistical aspects of associativity for copulas. (English) Zbl 1187.62094
Summary: We study in detail the associativity property of the discrete copulas. We observe the connection between discrete copulas and the empirical copulas, and then we propose a statistic that indicates when an empirical copula is associative and obtain its main statistical properties under independence. We also obtained asymptotic results of the proposed statistic. Finally, we study the associativity statistic under different copulas and we include some final remarks about associativity of samples.
##### MSC:
 62H05 Characterization and structure theory for multivariate probability distributions; copulas 62H20 Measures of association (correlation, canonical correlation, etc.) 60C05 Combinatorial probability
##### Keywords:
independence; discrete copulas; associativity; permutations
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##### References:
 [1] I. Aguiló, J. Suñer, and J. Torrens: Matrix representation of discrete quasi-copulas. Fuzzy Sets and Systems (2007), doi: 10.1016/j.fss2007.10.004. · Zbl 05599423 [2] C. Alsina, M. J. Frank, and B. Schweizer: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Co., Singapore 2006. · Zbl 1100.39023 [3] P. Deheuvels: La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. 65 (1979), 5, 274-292. · Zbl 0422.62037 [4] A. Erdely, J. M. González-Barrios, and R. B. Nelsen: Symmetries of random discrete copulas. Kybernetika 44 (2008), 6, 846-863. · Zbl 1206.62099 · www.kybernetika.cz · eudml:33969 [5] T. P. Hettmansperger: Statistical Inference Based on Ranks. Wiley, New York 1984. · Zbl 0665.62039 [6] S. Jenei: On the convex combination of left-continuous $$t$$-norms. Aequationes Mathematicae 72 (2006), 47-59. · Zbl 1101.39010 · doi:10.1007/s00010-006-2840-z [7] S. Jenei: On the geometry of associativity. Semigroup Forum 74 (2007), 439-466. · Zbl 1131.20047 · doi:10.1007/s00233-006-0673-7 [8] E. P. Klement, R. Mesiar, and E. Pap: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000. · Zbl 0972.03002 [9] E. P. Klement and R. Mesiar: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, Amsterdam 2005. · Zbl 1063.03003 [10] E. P. Klement and R. Mesiar: How non-symmetric can a copula be? Comment. Math. Univ. Carolinae 47 (2006), 1, 141-148. · Zbl 1150.62027 · emis:journals/CMUC/cmuc0601/cmuc0601.htm · eudml:22872 [11] A. Kolesárová, R. Mesiar, J. Mordelová, and C. Sempi: Discrete Copulas. IEEE Trans. Fuzzy Systems 14 (2006), 698-705. [12] A. Kolesárová and J. Mordelová: Quasi-copulas and copulas on a discrete scale. Soft Computing 10 (2006), 495-501. · Zbl 1096.60012 [13] E. L. Lehmann: Nonparametric Statistical Methods Based on Ranks. Revised first edition. Springer, New York 2006. [14] G. Mayor, J. Suñer, and J. Torrens: Copula-like operations on finite settings. IEEE Trans. Fuzzy Systems 13 (2005), 468-477. [15] G. Mayor, J. Suñer, and J. Torrens: Sklar’s Theorem in finite settings. IEEE Trans. Fuzzy Systems 15 (2007), 410-416. [16] R. Mesiar: Discrete copulas - what they are. Prof. Joint EUSFLAT-LFA 2005 (E. Montsenyand P. Sobrevilla Universitat Politecnica de Catalunya, Barcelona 2005, pp. 927-930. [17] R. B. Nelsen: An Introduction to Copulas. Second edition. Springer, New York 2006. · Zbl 1152.62030 [18] R. B. Nelsen: Extremes of nonexchangeability. Statist. Papers 48 (2007), 329-336. · Zbl 1110.62071 · doi:10.1007/s00362-006-0336-5 [19] B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North-Holland, New York 1983. · Zbl 0546.60010
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