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Keywords:
Archimedean copula; diagonal section; independence
Archimedean copula; diagonal section; independence
Summary:
In this paper we analyze some properties of the empirical diagonal and we obtain its exact distribution under independence for the two and three- dimensional cases, but the ideas proposed in this paper can be carried out to higher dimensions. The results obtained are useful in designing a nonparametric test for independence, and therefore giving solution to an open problem proposed by Alsina, Frank and Schweizer [2].
References:
[1] Aguiló I., Suñer, J., Torrens J.: Matrix representation of discrete quasi-copulas. Fuzzy Sets and Systems 159 (2008), 1658–1672 DOI 10.1016/j.fss.2007.10.004 | MR 2419976
[2] Alsina C., Frank M. J., Schweizer B.: Associative Functions: Triangular Norms and Copulas. World Scientific Publishing Co., Singapore 2006 MR 2222258 | Zbl 1100.39023
[3] Bailey D. F.: Counting arrangements of 1’s and $-1$’s. Math. Mag. 69 (1996), 128–131 MR 1573156 | Zbl 0859.05007
[4] Barcucci E., Verri M. C.: Some more properties of Catalan numbers. Discrete Math. 102 (1992), 229–237 DOI 10.1016/0012-365X(92)90117-X | MR 1169143 | Zbl 0757.05005
[5] Callan D.: Some bijections and identities for the Catalan and Fine numbers. Séminaire Lotharingen de Combinatoire 53 (2006), B53e (16 pp) MR 2221363 | Zbl 1084.05002
[6] Cameron N. T.: Random Walks, Trees and Extensions of Riordan Group Techniques. Ph.D. Thesis. Howard University 2002 MR 2703896
[7] Chung K. L., Feller W.: On fluctuations in coin-tossing. Proc. Nat. Acad. Sci. U. S. A. 35 (1949), 605–608 DOI 10.1073/pnas.35.10.605 | MR 0033459
[8] Darsow W. F., Frank M. J.: Associative functions and Abel–Schröder systems. Publ. Math. Debrecen 30 (1983), 253–272 MR 0739487 | Zbl 0542.39001
[9] Deheuvels P.: 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 MR 0573609 | Zbl 0422.62037
[10] Engleberg E. O.: On some problems concerning a restricted random walk. J. Appl. Probab. 2 (1965), 396–404 DOI 10.2307/3212201 | MR 0182993
[11] Erdely A.: Diagonal Properties of the Empirical Copula and Applications. Construction of Families of Copulas with Given Restrictions. Ph.D. Thesis, Universidad Nacional Autónoma de México 2007
[12] Feller W.: An Introduction to Probability Theory and Its Applications, Vol. 1. Third edition. Wiley, New York 1968 MR 0228020 | Zbl 0598.60003
[13] Frank M. J.: Diagonals of copulas and Schröder’s equation. Aequationes Math. 51 (1996), 150 MR 1144584
[14] Kimberling C. H.: A probabilistic interpretation of complete monotonicity. Aequationes Math. 10 (1974), 152–164 DOI 10.1007/BF01832852 | MR 0353416 | Zbl 0309.60012
[15] Klement E. P., Mesiar, R., Pap E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht 2000 MR 1790096 | Zbl 1087.20041
[16] Klement E. P., Mesiar R.: Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms. Elsevier, Amsterdam 2005 MR 2166082 | Zbl 1063.03003
[17] Kolesárová A., Mesiar R., Mordelová, J., Sempi C.: Discrete Copulas. IEEE Trans. Fuzzy Systems 14 (2006), 698–705 DOI 10.1109/TFUZZ.2006.880003
[18] Kolesárová A., Mordelová J.: Quasi-copulas and copulas on a discrete scale. Soft Computing 10 (2006), 495–501 DOI 10.1007/s00500-005-0524-6 | Zbl 1096.60012
[19] Kuczma M.: Functional Equations in a Single Variable. Polish Scientific Publishers, Warsaw 1968 MR 0228862 | Zbl 0725.39003
[20] Kuczma M., Choczewski, B., Ger R.: Iterative Functional Equations. Cambridge University Press, New York 1990 MR 1067720 | Zbl 1141.39023
[21] Mayor G., Suner, J., Torrens J.: Copula-like operations on finite settings. IEEE Trans. Fuzzy Systems 13 (2005), 468–477 DOI 10.1109/TFUZZ.2004.840129
[22] McNeil A. J., Nešlehová J.: Multivariate Archimedean copulas, $D$-monotone functions and $l_{1}$-norm symmetric distributions. Ann. Statist, to appear MR 2541455
[23] Mesiar R.: Discrete copulas – what they are. In: Joint EUSFLAT-LFA 2005, Conference Proc. (E. Montsenyand P. Sobrevilla, eds.) Universitat Politecnica de Catalunya, Barcelona 2005, pp. 927–930
[24] Nelsen R. B.: An Introduction to Copulas. Second edition. Springer, New York 2006 MR 2197664 | Zbl 1152.62030
[25] Schweizer B., Sklar A.: Probabilistic Metric Spaces. North-Holland, New York 1983 MR 0790314 | Zbl 0546.60010
[26] Sklar A.: Fonctions de répartition à $n$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229–231 MR 0125600
[27] Sungur E. A., Yang Y.: Diagonal copulas of Archimedean class. Comm. Statist. Theory Methods 25 (1996), 1659–1676 DOI 10.1080/03610929608831791 | MR 1411104 | Zbl 0900.62339
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