×

Some copula inference procedures adapted to the presence of ties. (English) Zbl 1464.62104

Summary: When modeling the distribution of a multivariate continuous random vector using the so-called copula approach, it is not uncommon to have ties in the coordinate samples of the available data because of rounding or lack of measurement precision. Yet, the vast majority of existing inference procedures on the underlying copula were both theoretically derived and practically implemented under the assumption of no ties. Applying them nonetheless can lead to strongly biased results. Some of the existing statistical tests can however be adapted to provide meaningful results in the presence of ties. It is the case of some tests of exchangeability, radial symmetry, extreme-value dependence and goodness of fit. Detailed algorithms for computing approximate \(p\)-values for the modified tests are provided and their finite-sample behaviors are empirically investigated through extensive Monte Carlo experiments. An illustration on a real-world insurance data set concludes the work.

MSC:

62-08 Computational methods for problems pertaining to statistics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G10 Nonparametric hypothesis testing
62G32 Statistics of extreme values; tail inference

Software:

copula; QRM; TwoCop; copula; R
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Agresti, A., Categorical data analysis, (2002), Wiley · Zbl 1018.62002
[2] Agresti, A., Analysis of ordinal categorical data, (2010), John Wiley and Sons New York · Zbl 1263.62007
[3] Beirlant, J.; Goegebeur, Y.; Segers, J.; Teugels, J., (Statistics of Extremes: Theory and Applications, Wiley Series in Probability and Statistics, (2004), John Wiley and Sons Ltd. Chichester)
[4] Ben Ghorbal, M.; Genest, C.; Nešlehová, J., On the test of ghoudi, khoudraji, and rivest for extreme-value dependence, Canad. J. Statist., 37, 4, 534-552, (2009) · Zbl 1191.62083
[5] Berg, D., Copula goodness-of-fit testing: an overview and power comparison, Eur. J. Finance, 15, 675-701, (2009)
[6] Berghaus, B.; Bücher, A., Goodness-of-fit tests for multivariate copula-based time series models, Econ. Theory, 33, 2, 292-330, (2017) · Zbl 1442.62730
[7] Berghaus, B.; Bücher, A.; Dette, H., Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence, J. Soc. Franç. Statist., 154, 1, 116-137, (2013) · Zbl 1316.62045
[8] Bücher, A.; Dette, H.; Volgushev, S., New estimators of the Pickands dependence function and a test for extreme-value dependence, Ann. Statist., 39, 4, 1963-2006, (2011) · Zbl 1306.62087
[9] Bücher, A.; Kojadinovic, I., An overview of nonparametric tests of extreme-value dependence and of some related statistical procedures, (Dey, D.; Yan, J., Extreme Value Modeling and Risk Analysis: Methods and Applications, (2015), Chapman and Hall/CRC), 377-398 · Zbl 1365.62174
[10] Capéraà, P.; Fougères, A.-L.; Genest, C., A nonparametric estimation procedure for bivariate extreme value copulas, Biometrika, 84, 567-577, (1997) · Zbl 1058.62516
[11] Cormier, E.; Genest, C.; Nešlehová, J. G., Using b-splines for nonparametric inference on bivariate extreme-value copulas, Extremes, 17, 633-659, (2014) · Zbl 1304.62072
[12] Deheuvels, P., La fonction de dépendance empirique et ses propriétés: un test non paramétrique d’indépendance, Acad. Roy. Belg. Bull. Cl. Sci. (5), 65, 274-292, (1979) · Zbl 0422.62037
[13] Deheuvels, P., A non parametric test for independence, Publ. Inst. Statist. Univ. Paris, 26, 29-50, (1981) · Zbl 0478.62029
[14] Fermanian, J.-D., An overview of the goodness-of-fit test problem for copulas, (Durante, F.; Jaworski, P.; Härdle, W., Copulae in Mathematical and Quantitative Finance, (2013), Springer), 61-89 · Zbl 1273.62101
[15] Frees, E. W.; Valdez, E. A., Understanding relationships using copulas, N. Am. Actuar. J., 2, 1-25, (1998) · Zbl 1081.62564
[16] Garralda-Guillem, A. I., Structure de dépendance des lois de valeurs extrêmes bivariées, C. R. Acad. Sci., Paris I, 330, 593-596, (2000) · Zbl 0951.60014
[17] Genest, C., Frank’s family of bivariate distributions, Biometrika, 74, 3, 549-555, (1987) · Zbl 0635.62038
[18] Genest, C.; Ghoudi, K.; Rivest, L.-P., A semiparametric estimation procedure of dependence parameters in multivariate families of distributions, Biometrika, 82, 543-552, (1995) · Zbl 0831.62030
[19] Genest, C.; Ghoudi, K.; Rivest, L.-P., Discussion of “understanding relationships using copulas”, by E. frees and E. valdez, N. Am. Actuar. J., 3, 143-149, (1998)
[20] Genest, C.; Huang, W.; Dufour, J.-M., A regularized goodness-of-fit test for copulas, J. Soc. Franç. Statist., 154, 64-77, (2013) · Zbl 1316.62075
[21] Genest, C.; Kojadinovic, I.; Nešlehová, J.; Yan, J., A goodness-of-fit test for bivariate extreme-value copulas, Bernoulli, 17, 1, 253-275, (2011) · Zbl 1284.62331
[22] Genest, C.; Nešlehová, J., A primer on copulas for count data, Astin Bull., 37, 475-515, (2007) · Zbl 1274.62398
[23] Genest, C.; Nešlehová, J. G., Assessing and modeling asymmetry in bivariate continuous data, (Jaworski, P.; Durante, F.; Härdle, W. K., Copulae in Mathematical and Quantitative Finance, Lecture Notes in Statistics, (2013), Springer), 91-114 · Zbl 1273.62112
[24] Genest, C.; Nešlehová, J. G., On tests of radial symmetry for bivariate copulas, Statist. Papers, 55, 1107-1119, (2014) · Zbl 1310.62069
[25] Genest, C.; Nešlehová, J. G.; Rémillard, B., On the empirical multilinear copula process for count data, Bernoulli, 20, 1344-1371, (2014) · Zbl 1365.62221
[26] Genest, C.; Nešlehová, J.; Ruppert, M., Comment on the paper by S. haug, C. klüppelberg and L. peng entitled statistical models and methods for dependence in insurance data, J. Korean Stat. Soc., 40, 141-148, (2011)
[27] Genest, C.; Nešlehová, J.; Quessy, J.-F., Tests of symmetry for bivariate copulas, Ann. Inst. Statist. Math., 64, 811-834, (2012) · Zbl 1440.62182
[28] Genest, C.; Rémillard, B., Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models, Ann. Inst. H. Poincaré Probab. Statist., 44, 1096-1127, (2008) · Zbl 1206.62044
[29] Genest, C.; Rémillard, B.; Beaudoin, D., Goodness-of-fit tests for copulas: A review and a power study, Insurance Math. Econom., 44, 199-213, (2009) · Zbl 1161.91416
[30] Genest, C.; Rivest, L.-P., Statistical inference procedures for bivariate Archimedean copulas, J. Amer. Statist. Assoc., 88, 423, 1034-1043, (1993) · Zbl 0785.62032
[31] Genest, C.; Segers, J., Rank-based inference for bivariate extreme-value copulas, Ann. Statist., 37, 2990-3022, (2009) · Zbl 1173.62013
[32] Ghoudi, K.; Khoudraji, A.; Rivest, L.-P., Propriétés statistiques des copules de valeurs extrêmes bidimensionnelles, Canad. J. Statist., 26, 1, 187-197, (1998) · Zbl 0899.62071
[33] Gudendorf, G., Nonparametric estimation of multivariate extreme-value copulas, (2012), Université catholique de Louvain, (Ph.D. thesis) · Zbl 1349.62207
[34] Gudendorf, G.; Segers, J., Extreme-value copulas, (Jaworski, P.; Durante, F.; Härdle, W.; Rychlik, W., Copula Theory and its Applications (Warsaw, 2009), Lecture Notes in Statistics, (2010), Springer-Verlag), 127-146
[35] Gumbel, E. J., Statistics of extremes, (1958), Columbia University Press New York · Zbl 0086.34401
[36] Hofert, M., Kojadinovic, I., Mächler, M., Yan, J., 2017. copula: Multivariate dependence with copulas. URL http://CRAN.R-project.org/package=copula. R package version 0.999-17.
[37] Kendall, M. G., The treatment of ties in rank problems, Biometrika, 3, 239-251, (1945) · Zbl 0063.03216
[38] Khoudraji, A., Contributions à l’étude des copules et à la modélisation des valeurs extrêmes bivariées, (1995), Université Laval Québec, Canada, (Ph.D. thesis)
[39] Kojadinovic, I.; Segers, J.; Yan, J., Large-sample tests of extreme-value dependence for multivariate copulas, Canad. J. Statist., 39, 4, 703-720, (2011) · Zbl 1284.62333
[40] Kojadinovic, I.; Yan, J., Modeling multivariate distributions with continuous margins using the R package, J. Stat. Softw., 34, 9, 1-20, (2010)
[41] Kojadinovic, I.; Yan, J., Nonparametric rank-based tests of bivariate extreme-value dependence, J. Multivariate Anal., 101, 9, 2234-2249, (2010) · Zbl 1201.62056
[42] Kojadinovic, I.; Yan, J., A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems, Stat. Comput., 21, 1, 17-30, (2011) · Zbl 1274.62400
[43] Kojadinovic, I.; Yan, J., A nonparametric test of exchangeability for extreme-value and left-tail decreasing bivariate copulas, Scand. J. Statist., 39, 3, 480-496, (2012) · Zbl 1323.62035
[44] Kojadinovic, I.; Yan, J.; Holmes, M., Fast large-sample goodness-of-fit for copulas, Statist. Sinica, 21, 2, 841-871, (2011) · Zbl 1214.62049
[45] Liebscher, E., Construction of asymmetric multivariate copulas, J. Multivariate Anal., 99, 2234-2250, (2008) · Zbl 1151.62043
[46] McNeil, A. J.; Frey, R.; Embrechts, P., Quantitative risk management: concepts, techniques and tools, (2015), Princeton University Press · Zbl 1337.91003
[47] Nelsen, R. B., An introduction to copulas, (2006), Springer New York · Zbl 1152.62030
[48] Oakes, D., A model for association in bivariate survival data, J. R. Stat. Soc. Ser. B Stat. Methodol., 44, 414-422, (1982) · Zbl 0503.62035
[49] Pappadà, R.; Durante, F.; Salvadori, G., Quantification of the environmental structural risk with spoiling ties: Is randomization worthwhile?, Stoch. Environ. Res. Risk Assess., (2016), URL http://dx.doi.org/10.1007/s00477-016-1357-9
[50] Patton, A. J., Copula methods for forecasting multivariate time series, (Handbook of Economic Forecasting, Vol. 2, (2012), Springer Verlag)
[51] Pickands, J., Multivariate extreme value distributions. with a discussion, Bull. Inst. Internat. Statist., 49, 859-878, (1981), 894-902. Proceedings of the 43rd session of the Internatinal Statistical Institute · Zbl 0518.62045
[52] R Core Team, 2016. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org.
[53] Rémillard, B.; Scaillet, O., Testing for equality between two copulas, J. Multivariate Anal., 100, 3, 377-386, (2009) · Zbl 1157.62401
[54] Rüschendorf, L., On the distributional transform, sklar’s theorem, and the empirical copula process, J. Statist. Plann. Inference, 139, 11, 3921-3927, (2009) · Zbl 1171.60313
[55] Salvadori, G.; De Michele, C.; Kottegoda, N. T.; Rosso, R., (Extremes in Nature: An Approach Using Copulas, Water Science and Technology Library, vol. 56, (2007), Springer)
[56] Sklar, A., Fonctions de répartition à \(n\) dimensions et leurs marges, Publ. Inst. Statist. Univ. Paris, 8, 229-231, (1959) · Zbl 0100.14202
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.