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A goodness-of-fit test for multivariate multiparameter copulas based on multiplier central limit theorems. (English) Zbl 1274.62400
Summary: Recent large scale simulations indicate that a powerful goodness-of-fit test for copulas can be obtained from the process comparing the empirical copula with a parametric estimate of the copula derived under the null hypothesis. A first way to compute approximate $$p$$-values for statistics derived from this process consists of using the parametric bootstrap procedure recently thoroughly revisited by C. Genest and B. Rémillard [Ann. Inst. Henri Poincaré, Probab. Stat. 44, No. 6, 1096–1127 (2008; Zbl 1206.62044)]. Because it heavily relies on random number generation and estimation, the resulting goodness-of-fit test has a very high computational cost that can be regarded as an obstacle to its application as the sample size increases. An alternative approach proposed by the authors consists of using a multiplier procedure. The study of the finite-sample performance of the multiplier version of the goodness-of-fit test for bivariate one-parameter copulas showed that it provides a valid alternative to the parametric bootstrap-based test while being orders of magnitude faster. The aim of this work is to extend the multiplier approach to multivariate multiparameter copulas and study the finite-sample performance of the resulting test. Particular emphasis is put on elliptical copulas such as the normal and the $$t$$ as these are flexible models in a multivariate setting. The implementation of the procedure for the latter copulas proves challenging and requires the extension of the Plackett formula for the $$t$$-distribution to arbitrary dimension. Extensive Monte Carlo experiments, which could be carried out only because of the good computational properties of the multiplier approach, confirm in the multivariate multiparameter context the satisfactory behavior of the goodness-of-fit test.

##### MSC:
 62H20 Measures of association (correlation, canonical correlation, etc.) 60F05 Central limit and other weak theorems
##### Software:
copula; R; copula ; mvtnorm; TwoCop; QSIMVN
Full Text:
##### References:
 [1] Berg, D.: Copula goodness-of-fit testing: An overview and power comparison. Eur. J. Finance (2009, in press) [2] Berg, D., Quessy, J.-F.: Local sensitivity analyses of goodness-of-fit tests for copulas. Scand. J. Stat. (2009, in press) [3] Charpentier, A., Fermanian, J.-D., Scaillet, O.: The estimation of copulas: Theory and practice. In: Rank, J. (ed.) Copulas: From Theory to Application in Finance, pp. 35–60. Risk Books, New York (2007) [4] Cherubini, G., Vecchiato, W., Luciano, E.: Copula Models in Finance, 2nd edn. The Wiley Finance Series. Wiley, New York (2004) · Zbl 1163.62081 [5] Cornish, E.A.: The multivariate t-distribution associated with a set of normal sample deviates. Aust. J. Phys. 7, 531–542 (1954) · Zbl 0059.13201 [6] Cui, S., Sun, Y.: Checking for the gamma frailty distribution under the marginal proportional hazards frailty model. Stat. Sin. 14, 249–267 (2004) · Zbl 1054.62118 [7] Deheuvels, P.: A non parametric test for independence. Publ. Inst. Stat. Univ. Paris 26, 29–50 (1981) · Zbl 0478.62029 [8] Demarta, S., McNeil, A.: The t copula and related copulas. Int. Stat. Rev. 73(1), 111–129 (2005) · Zbl 1104.62060 [9] Frees, E.W., Valdez, E.A.: Understanding relationships using copulas. North Am. Actuar. J. 2, 1–25 (1998) · Zbl 1081.62564 [10] Genest, C., Favre, A.-C.: Everything you always wanted to know about copula modeling but were afraid to ask. J. Hydrol. Eng. 12, 347–368 (2007) [11] Genest, C., Rémillard, B.: Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann. Inst. Henri Poincaré: Probab. Stat. 44, 1096–1127 (2008) · Zbl 1206.62044 [12] Genest, C., Werker, B.J.M.: Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models. In: Cuadras, C.M., Fortiana, J., Rodríguez Lallena, J.A. (eds.) Distributions with Given Marginals and Statistical Modelling, pp. 103–112. Kluwer, Dordrecht (2002) · Zbl 1142.62330 [13] 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 [14] Genest, C., Favre, A.-C., Béliveau, J., Jacques, C.: Metaelliptical copulas and their use in frequency analysis of multivariate hydrological data. Water Resour. Res. 43, 12 (2007) [15] Genest, C., Rémillard, B., Beaudoin, D.: Goodness-of-fit tests for copulas: A review and a power study. Insur. Math. Econ. 44, 199–213 (2009) · Zbl 1161.91416 [16] Genz, A.: Numerical computation of multivariate normal probabilities. J. Comput. Graph. Stat. 1, 141–150 (1992) [17] Genz, A.: Comparison of methods for the computation of multivariate normal probabilities. Comput. Sci. Stat. 25, 400–405 (1993) [18] Genz, A.: Numerical computation of rectangular bivariate and trivariate normal and t probabilities. Stat. Comput. 14, 251–260 (2004) [19] Genz, A., Bretz, F.: Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. J. Stat. Comput. Simul. 63, 361–378 (1999) · Zbl 0934.62020 [20] Genz, A., Bretz, F.: Methods for the computation of multivariate t-probabilities. J. Comput. Graph. Stat. 11, 950–971 (2002) [21] Genz, A., Bretz, F., Hothorn, T.: mvtnorm: Multivariate normal and t distribution (2007). R package version 0.8-1 [22] Joe, H.: Multivariate Models and Dependence Concepts. Chapman and Hall, London (1997) · Zbl 0990.62517 [23] Kim, G., Silvapulle, M.J., Silvapulle, P.: Comparison of semiparametric and parametric methods for estimating copulas. Comput. Stat. Data Anal. 51(6), 2836–2850 (2007) · Zbl 1161.62364 [24] Kojadinovic, I., Yan, J.: Fast large-sample goodness-of-fit for copulas. Technical Report 24, Department of Statistics, The University of Connecticut (2009) · Zbl 1214.62049 [25] McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management. Princeton Series in Finance. Princeton University Press, Princeton (2005) · Zbl 1089.91037 [26] Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006) · Zbl 1152.62030 [27] Plackett, R.L.: A reduction formula for normal multivariate probabilities. Biometrika 41, 351–369 (1954) · Zbl 0056.35702 [28] Quessy, J.-F.: Méthodologie et application des copules: tests d’adéquation, tests d’indépendance, et bornes sur la valeur-à-risque. PhD thesis, Université Laval, Québec, Canada (2005) [29] R Development Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2009). http://www.R-project.org . ISBN 3-900051-07-0 [30] Rémillard, B., Scaillet, O.: Testing for equality between two copulas. J. Multivar. Anal. 100(3), 377–386 (2009) · Zbl 1157.62401 [31] Ruymgaart, F.H., Shorack, G.R., van Zwet, W.R.: Asymptotic normality of nonparametric tests for independence. Ann. Math. Stat. 43(4), 1122–1135 (1972) · Zbl 0254.62025 [32] Scaillet, O.: A Kolmogorov-Smirnov type test for positive quadrant dependence. Can. J. Stat. 33, 415–427 (2005) · Zbl 1077.62036 [33] Seber, G.A.F.: A Matrix Handbook for Statisticians. Wiley Series in Probability and Statistics. Wiley, New York (2008) [34] Shih, J.H., Louis, T.A.: Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51(4), 1384–1399 (1995) · Zbl 0869.62083 [35] Sklar, A.: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris 8, 229–231 (1959) · Zbl 0100.14202 [36] Stute, W., González, W., Presedo-Quindimil, M.: Bootstrap based goodness-of-fit tests. Metrika 40, 243–256 (1993) · Zbl 0770.62016 [37] Yan, J., Kojadinovic, I.: copula: Multivariate dependence with copulas (2008). R package version 0.8-6
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