Asymptotic total variation tests for copulas.(English)Zbl 1331.62282

This work introduces a new goodness-of-fit test for copulas, based on empirical copula processes and nonparametric bootstrap counterparts. The new test extends the standard Kolmogorov-Smirnov type test for copulas in the sense that while the Kolmogorov-Smirnov test takes the supremum of the empirical copula process indexed by orthants, the new test is based on the empirical copula process indexed by families of $$L_n$$ disjoint boxes with $$L_n$$ tending to infinity at an adequate slower rate than $$n.$$ At such rate the underlying empirical process does not converge. However, the critical values of the new test statistics can be consistently estimated by their nonparametric bootstrap counterparts. It is shown else that for parametric hypothesis, i.e., when the copula is indexed by a parameter, the new test works well. Some applications are presented and simulations show that the power of the new test is consistently higher than the power of the standard Kolmogorov-Smirnov or the Cramér-von Mises test for copula on the examples worked in the paper.

MSC:

 62H15 Hypothesis testing in multivariate analysis 62G09 Nonparametric statistical resampling methods 62G10 Nonparametric hypothesis testing

TwoCop
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 [1] Appel, M.J., LaBarre, R. and Radulović, D. (2003). On accelerated random search. SIAM J. Optim. 14 708-731 (electronic). · Zbl 1061.65046 [2] Berg, D. (2009). Copula goodness-of-fit testing: An overview and power comparison. Eur. J. Finance 15 675-701. [3] Bickel, P.J. and Freedman, D.A. (1981). Some asymptotic theory for the bootstrap. Ann. Statist. 9 1196-1217. · Zbl 0449.62034 [4] Bickel, P.J. and Freedman, D.A. (1983). Bootstrapping regression models with many parameters. In A Festschrift for Erich L. Lehmann (P.J. Bickel, K. Doksum and J.L. Hodges, eds.). Wadsworth Statist./Probab. Ser. 28-48. Belmont, CA: Wadsworth. · Zbl 0529.62057 [5] Bücher, A. and Dette, H. (2010). A note on bootstrap approximations for the empirical copula process. Statist. Probab. Lett. 80 1925-1932. · Zbl 1202.62055 [6] Csörgó, S. and Mason, D.M. (1989). Bootstrapping empirical functions. Ann. Statist. 17 1447-1471. · Zbl 0701.62057 [7] Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés. Un test non paramétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. (5) 65 274-292. · Zbl 0422.62037 [8] Dobrić, J. and Schmid, F. (2005). Testing goodness of fit for parametric families of copulas - Application to financial data. Comm. Statist. Simulation Comput. 34 1053-1068. · Zbl 1080.62040 [9] Fermanian, J.-D. (2005). Goodness-of-fit tests for copulas. J. Multivariate Anal. 95 119-152. · Zbl 1095.62052 [10] Fermanian, J.-D. (2012). An overview of the goodness-of-fit test problem for copulas. In Copulae in Mathematical and Quantitative Finance (P. Jaworski, F. Durant and W. Härdle, eds.) 68-89. Berlin: Springer. · Zbl 1273.62101 [11] Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847-860. · Zbl 1068.62059 [12] Fermanian, J.-D. and Wegkamp, M.H. (2012). Time-dependent copulas. J. Multivariate Anal. 110 19-29. · Zbl 1352.62073 [13] Genest, C., Ghoudi, K. and Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika 82 543-552. · Zbl 0831.62030 [14] Genest, C. and Rémillard, B. (2008). Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann. Inst. Henri Poincaré Probab. Stat. 44 1096-1127. · Zbl 1206.62044 [15] Genest, C., Rémillard, B. and Beaudoin, D. (2009). Goodness-of-fit tests for copulas: A review and a power study. Insurance Math. Econom. 44 199-213. · Zbl 1161.91416 [16] Hildebrandt, T.H. (1963). Introduction to the Theory of Integration. Pure and Applied Mathematics XIII . New York: Academic Press. [17] Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19 293-325. · Zbl 0032.04101 [18] Hvattum, L.M. and Glover, F. (2009). Finding local optima of high-dimensional functions using direct search methods. European J. Oper. Res. 195 31-45. · Zbl 1156.90440 [19] Liebscher, E. (2008). Construction of asymmetric multivariate copulas. J. Multivariate Anal. 99 2234-2250. · Zbl 1151.62043 [20] Mason, D.M. (1981). Bounds for weighted empirical distribution functions. Ann. Probab. 9 881-884. · Zbl 0478.60036 [21] Nelsen, R.B. (2006). An Introduction to Copulas , 2nd ed. New York: Springer. · Zbl 1152.62030 [22] Radulović, D. (1998). Can we bootstrap even if CLT fails? J. Theoret. Probab. 11 813-830. · Zbl 0979.62025 [23] Radulović, D. (2012). A direct bootstrapping technique and its application to a novel goodness of fit test. J. Multivariate Anal. 107 181-199. · Zbl 1238.62051 [24] Radulović, D. (2013). High dimensional CLT and its applications. In High Dimensional Probability VI. Progress in Probability 66 357-373. Basel: Springer. · Zbl 1271.60040 [25] Rémillard, B. and Scaillet, O. (2009). Testing for equality between two copulas. J. Multivariate Anal. 100 377-386. · Zbl 1157.62401 [26] Scaillet, O. (2007). Kernel-based goodness-of-fit tests for copulas with fixed smoothing parameters. J. Multivariate Anal. 98 533-543. · Zbl 1107.62037 [27] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764-782. · Zbl 1243.62066 [28] Shih, J.H. and Louis, T.A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics 51 1384-1399. · Zbl 0869.62083 [29] Shorack, G.R. and Wellner, J.A. (2009). Empirical Processes with Applications to Statistics , 2nd ed. Philadelphia, PA: SIAM. · Zbl 1171.62057 [30] Sklar, M. (1959). Fonctions de répartition à $$n$$ dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 229-231. · Zbl 0100.14202 [31] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3 . Cambridge: Cambridge Univ. Press. · Zbl 0910.62001 [32] van der Vaart, A.W. and Wellner, J. (2000). Weak Convergence and Empirical Processes. Springer Series in Statistics . New York: Springer.
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