×

A test for Archimedeanity in bivariate copula models. (English) Zbl 1243.62078

Summary: We propose a new test for the hypothesis that a bivariate copula is an Archimedean copula which can be used as a preliminary step before further dependence modeling. The corresponding test statistic is based on a combination of two measures resulting from the characterization of Archimedean copulas by the property of associativity and by a strict upper bound on the diagonal by the Fréchet-Hoeffding upper bound. We prove weak convergence of this statistic and show that the critical values of the corresponding test can be determined by the multiplier bootstrap method. The test is shown to be consistent against all departures from Archimedeanity. A simulation study is presented which illustrates the finite-sample properties of the new test.

MSC:

62H15 Hypothesis testing in multivariate analysis
62H20 Measures of association (correlation, canonical correlation, etc.)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60F05 Central limit and other weak theorems
62G09 Nonparametric statistical resampling methods
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

Software:

TwoCop
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alsina, C.; Frank, M. J.; Schweizer, B., Associative Functions—Triangular Norms and Copulas (2006), World Scientific Publishing Co. Pvt. Ltd.: World Scientific Publishing Co. Pvt. Ltd. Hackensack, NJ · Zbl 1100.39023
[2] Ben Ghorbal, N.; Genest, C.; Nešlehová, J., On the Ghoudi, Khoudraji, and Rivest test for extreme-value dependence, Canad. J. Statist., 37, 534-552 (2009) · Zbl 1191.62083
[3] A. Bücher, Statistical inference for copulas and extremes, Ph.D. Thesis, Ruhr-Universität Bochum, Germany, 2011.; A. Bücher, Statistical inference for copulas and extremes, Ph.D. Thesis, Ruhr-Universität Bochum, Germany, 2011.
[4] Bücher, A.; Dette, H., A note on bootstrap approximations for the empirical copula process, Statist. Probab. Lett., 80, 1925-1932 (2010) · Zbl 1202.62055
[5] Charpentier, A.; Segers, J., Tails of multivariate Archimedean copulas, J. Multivariate Anal., 100, 1521-1537 (2009) · Zbl 1165.62038
[6] Cook, D. R.; Johnson, M. E., Generalized Burr-Pareto-logistic distributions with applications to a uranium exploration data set, Technometrics, 28, 123-131 (1986)
[7] Fermanian, J.-D.; Radulović, D.; Wegkamp, M., Weak convergence of empirical copula processes, Bernoulli, 10, 847-860 (2004) · Zbl 1068.62059
[8] Gänssler, P.; Stute, W., (Seminar on Empirical Processes. Seminar on Empirical Processes, DMV Seminar, vol. 9 (1987), Birkhäuser Verlag: Birkhäuser Verlag Basel) · Zbl 0637.62047
[9] Genest, C.; MacKay, R. J., Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données, Canad. J. Statist., 14, 145-159 (1986) · Zbl 0605.62049
[10] Genest, C.; Nešlehová, J.; Ziegel, J., Inference in multivariate Archimedean copula models, TEST, 20, 223-292 (2011) · Zbl 1274.62399
[11] Genest, C.; Quessy, J.-F.; Rémillard, B., Goodness-of-fit procedures for copula models based on the probability integral transformation, Scandinavian J. Statist., 33, 337-366 (2006) · Zbl 1124.62028
[12] Genest, C.; Rivest, L.-P., A characterization of Gumbel’s family of extreme value distributions, Statist. Probab. Lett., 8, 207-211 (1989) · Zbl 0701.62060
[13] Genest, C.; Rivest, L.-P., Statistical inference procedures for bivariate Archimedean copulas, J. Amer. Statist. Assoc., 88, 1034-1043 (1993) · Zbl 0785.62032
[14] Genest, C.; Segers, J., Rank-based inference for bivariate extreme-value copulas, Ann. Statist., 37, 2990-3022 (2009) · Zbl 1173.62013
[15] Jaworski, P., Testing archimedeanity, (Borgelt, C.; González-Rodríguez, G.; Trutschnig, W.; Lubiano, M.; Gil, M.; Grzegorzewski, P.; Hryniewicz, O., Combining Soft Computing and Statistical Methods in Data Analysis. Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, vol. 77 (2010), Springer: Springer Berlin), 353-360
[16] Joe, H., Families of min-stable multivariate exponential and multivariate extreme value distributions, Statist. Probab. Lett., 9, 75-81 (1990) · Zbl 0686.62035
[17] Kosorok, M. R., Introduction to Empirical Processes and Semiparametric Inference (2008), Springer: Springer New York · Zbl 1180.62137
[18] Larsson, M.; Nešlehová, J., Extremal behavior of Archimedean copulas, Adv. Appl. Probab., 43, 195-216 (2011) · Zbl 1213.62084
[19] Ling, C.-H., Representation of associative functions, Publ. Math. Debrecen, 12, 189-212 (1965) · Zbl 0137.26401
[20] McNeil, A. J.; Nešlehová, J., Multivariate Archimedean copulas, \(d\)-monotone functions and \(\ell_1\)-norm symmetric distributions, Ann. Statist., 37, 3059-3097 (2009) · Zbl 1173.62044
[21] Naifar, N., Modelling dependence structure with Archimedean copulas and applications to the iTraxx CDS index, J. Comput. Appl. Math., 235, 2459-2466 (2011) · Zbl 1208.91163
[22] Nelsen, R. B., (An Introduction to Copulas. An Introduction to Copulas, Springer Series in Statistics (2006), Springer: Springer New York) · Zbl 1152.62030
[23] J.-F. Quessy, Testing for bivariate extreme dependence using Kendall’s process, Scand. J. Statist. (2011), in press (doi:10.1111/j.1467-9469.2011.00739.x; J.-F. Quessy, Testing for bivariate extreme dependence using Kendall’s process, Scand. J. Statist. (2011), in press (doi:10.1111/j.1467-9469.2011.00739.x
[24] Rémillard, B.; Scaillet, O., Testing for equality between two copulas, J. Multivariate Anal., 100, 377-386 (2009) · Zbl 1157.62401
[25] Rivest, L.-P.; Wells, M. T., A martingale approach to the copula-graphic estimator for the survival function under dependent censoring, J. Multivariate Anal., 79, 138-155 (2001) · Zbl 1027.62069
[26] Rüschendorf, L., Asymptotic distributions of multivariate rank order statistics, Ann. Statist., 4, 912-923 (1976) · Zbl 0359.62040
[27] Scaillet, O., A Kolmogorov-Smirnov type test for positive quadrant dependence, Canad. J. Statist., 33, 415-427 (2005) · Zbl 1077.62036
[28] J. Segers, Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions, 2010. arXiv:1012.2133v1; J. Segers, Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions, 2010. arXiv:1012.2133v1
[29] J. Segers, Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions, 2011. arXiv:1012.2133v2; J. Segers, Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions, 2011. arXiv:1012.2133v2
[30] Sklar, A., Fonctions de répartition à \(n\) dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231 (1959) · Zbl 0100.14202
[31] Tsukahara, H., Semiparametric estimation in copula models, Canad. J. Statist., 33, 357-375 (2005) · Zbl 1077.62022
[32] van der Vaart, A. W.; Wellner, J. A., (Weak Convergence and Empirical Processes. Weak Convergence and Empirical Processes, Springer Series in Statistics (1996), Springer: Springer New York) · Zbl 0862.60002
[33] Wang, W.; Wells, M. T., Model selection and semiparametric inference for bivariate failure-time data, J. Amer. Statist. Assoc., 95, 62-72 (2000) · Zbl 0996.62091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.