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A Journal of Theoretical and Applied Statistics
Volume 47, 2013 - Issue 5
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Original Articles

Strong approximation of empirical copula processes by Gaussian processes

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Pages 1047-1063 | Received 29 Mar 2010, Accepted 22 Mar 2012, Published online: 16 May 2012

References

  • Sklar , A. 1959 . Fonctions de répartition à n dimensions et leurs marges . Publ. Inst. Statist. Univ. Paris , 8 : 229 – 231 .
  • Nelsen , R. B. 2006 . An Introduction to Copulas , 2 , Springer Series in Statistics New York : Springer .
  • Joe , H. 1997 . “ Multivariate Models and Dependence Concepts ” . In Monographs on Statistics and Applied Probability Vol. 73 , London : Chapman & Hall .
  • Schweizer , B. 1991 . “ Thirty years of copulas ” . In Advances in Probability Distributions with Given Marginals , Edited by: Dall'Aglio , G. , Kotz , S. and Salinetti , G. 13 – 50 . Dordrecht : Kluwer Academic Publishers . Mathematics and its Applications Vol. 67
  • Frees , E. W. and Valdez , E. A. 1998 . Understanding relationships using copulas . N. Am. Actuar. J. , 2 ( 1 ) : 1 – 25 . (doi:10.1080/10920277.1998.10595667)
  • Cui , S. and Sun , Y. 2004 . Checking for the gamma frailty distribution under the marginal proportional hazards frailty model . Statist. Sinica , 14 ( 1 ) : 249 – 267 .
  • Cherubini , U. , Luciano , E. and Vecchiato , W. 2004 . “ Copula Methods in Finance ” . Wiley Finance Series Chichester : Wiley .
  • McNeil , A. J. , Frey , R. and Embrechts , P. 2005 . “ Quantitative risk management, Concepts, techniques and tools ” . Princeton Series in Finance Princeton , NJ : Princeton University Press .
  • 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 .
  • Sklar , A. 1973 . Random variables, joint distribution functions, and copulas . Kybernetika (Prague) , 9 : 449 – 460 .
  • Philipp , W. and Pinzur , L. 1980 . Almost sure approximation theorems for the multivariate empirical process . Z. Wahrsch. Verw. Gebiete , 54 ( 1 ) : 1 – 13 . (doi:10.1007/BF00535346)
  • Wichura , M. J. 1973 . Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments . Ann. Probab. , 1 : 272 – 296 . (doi:10.1214/aop/1176996980)
  • Moore , D. S. and Spruill , M. C. 1975 . Unified large-sample theory of general chi-squared statistics for tests of fit . Ann. Statist. , 3 : 599 – 616 .
  • Ruymgaart , F. 1973 . “ Asymptotic theory for rank tests for independence, MC Tract 43 ” . Amsterdam : Mathematisch Institut . Ph.D. thesis
  • Stute , W. 1984 . The oscillation behavior of empirical processes: The multivariate case . Ann. Probab. , 12 : 361 – 379 . (doi:10.1214/aop/1176993295)
  • Gaenssler , P. and Stute , W. 1987 . “ Seminar on Empirical Processes 9. DMV Seminar ” . Basel : Birkhäuser Verlag .
  • Rüschendorf , L. 1974 . On the empirical process of multivariate, dependent random variables . J. Multivariate Anal. , 4 : 469 – 478 . (doi:10.1016/0047-259X(74)90025-6)
  • Rüschendorf , L. 1976 . Asymptotic distributions of multivariate rank order statistics . Ann. Statist. , 4 ( 5 ) : 912 – 923 . (doi:10.1214/aos/1176343588)
  • Tsukahara , H. 2005 . Semiparametric estimation in copula models . Canad. J. Statist. , 33 ( 3 ) : 357 – 375 . (doi:10.1002/cjs.5540330304)
  • Rüschendorf , L. 2009 . On the distributional transform, Sklar's theorem, and the empirical copula process . J. Statist. Plann. Inference , 139 ( 11 ) : 3921 – 3927 . (doi:10.1016/j.jspi.2009.05.030)
  • Deheuvels , P. 1980 . “ Nonparametric test of independence ” . In Nonparametric Asymptotic Statistics (Proc. Conf., Rouen, 1979) (French) , Edited by: Raoult , J. P. 95 – 107 . Berlin : Springer . Lecture Notes in Mathematics Vol. 821
  • Deheuvels , P. 1981 . “ Multivariate tests of independence ” . In Analytical Methods in Probability Theory , Edited by: Dugué , D. , Lukacs , E. and Rohatgi , V. K. 42 – 50 . Berlin : Springer . Lecture Notes in Mathematics Vol. 861
  • Fermanian , J.-D. , Radulović , D. and Wegkamp , M. 2004 . Weak convergence of empirical copula processes . Bernoulli , 10 ( 5 ) : 847 – 860 . (doi:10.3150/bj/1099579158)
  • Segers , J. 2012 . Weak convergence of empirical copula processes under nonrestrictive smoothness assumptions . Bernoulli , 18 (in press) (doi:10.3150/11-BEJ387)
  • Komlós , J. , Major , P. and Tusnády , G. 1975 . An approximation of partial sums of independent RV's and the sample DF. I . Z. Wahrscheinlichkeitstheorie und Verw. Gebiete , 32 : 111 – 131 . (doi:10.1007/BF00533093)
  • DasGupta , A. 2008 . “ Asymptotic Theory of Statistics and Probability ” . Springer Texts in Statistics New York : Springer .
  • Csörgő , M. and Horváth , L. 1993 . “ Weighted Approximations in Probability and Statistics ” . Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics Chichester : Wiley .
  • Csörgő , M. and Révész , P. 1981 . “ Strong Approximations in Probability and Statistics ” . New York : Probability and Mathematical Statistics, Academic Press Inc., Harcourt Brace Jovanovich Publishers .
  • Shorack , G. R. and Wellner , J. A. 1986 . “ Empirical Processes with Applications to Statistics ” . Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics New York : Wiley .
  • Csörgő , S. and Hall , P. 1984 . The Komlós-Major-Tusnády approximations and their applications . Austral. J. Statist. , 26 ( 2 ) : 189 – 218 . (doi:10.1111/j.1467-842X.1984.tb01233.x)
  • Deheuvels , P. 2009 . A multivariate Bahadur-Kiefer representation for the empirical copula process . Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , 364 : 120 – 147 . (Veroyatnost i Statistika. 14.2)
  • Piterbarg , V. I. 1996 . “ Asymptotic Methods in the Theory of Gaussian Processes and Fields ” . In Translations of Mathematical Monographs Vol. 148 , Providence , RI : American Mathematical Society . Translated from the Russian by V.V. Piterbarg, revised by the author
  • Adler , R. J. 1990 . “ An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes ” . Institute of Mathematical Statistics Lecture Notes – Monograph Series 12 Hayward , CA : Institute of Mathematical Statistics .
  • Berkes , I. and Philipp , W. 1979 . Approximation theorems for independent and weakly dependent random vectors . Ann. Probab. , 7 ( 1 ) : 29 – 54 . (doi:10.1214/aop/1176995146)
  • Csörgő , M. 1979 . Strong approximations of the Hoeffding, Blum, Kiefer, Rosenblatt multivariate empirical process . J. Multivariate Anal. , 9 ( 1 ) : 84 – 100 . (doi:10.1016/0047-259X(79)90068-X)
  • Tsukahara , H. 2000 . “ Empirical copulas and some applications ” . In Research Report 27 , The Institute for Economic Studies, Seijo University .
  • Bouzebda , S. , Keziou , A. and Zari , T. 2011 . K-sample problem using strong approximations of empirical copula processes . Math. Methods Statist. , 20 ( 2 ) : 14 – 29 . (doi:10.3103/S1066530711010029)
  • Bouzebda , S. and El Faouzi , N.-E. 2012 . New two-sample tests based on the integrated empirical copula processes . Statistics , DOI: 10.1080/02331888.2010.512976
  • Bouzebda , S. , El Faouzi , N.-E. and Zari , T. 2011 . On the multivariate two-sample problem using strong approximations of empirical copula processes . Comm. Statist. Theory Methods , 40 ( 8 ) : 1490 – 1509 . (doi:10.1080/03610921003615856)
  • Rémillard , B. “ Goodness-of-Fit Tests for Copulas of Multivariate Time Series (December 22, 2010) ” . Available at http://ssrn.com/abstract=1729982
  • Csörgő , M. , Horváth , L. and Szyszkowicz , B. 1997 . Integral tests for suprema of Kiefer processes with application . Statist. Decis. , 15 ( 4 ) : 365 – 377 .
  • van der Vaart , A. W. and Wellner , J. A. 1996 . “ Weak convergence and empirical processes ” . Springer Series in Statistics New York : Springer-Verlag .
  • Rémillard , B. and Scaillet , O. 2009 . Testing for equality between two copulas . J. Multivariate Anal. , 100 ( 3 ) : 377 – 386 . (doi:10.1016/j.jmva.2008.05.004)
  • Scaillet , O. 2005 . A Kolmogorov-Smirnov type test for positive quadrant dependence . Canad. J. Statist. , 33 ( 3 ) : 415 – 427 . (doi:10.1002/cjs.5540330307)
  • Ghoudi , K. and Rémillard , B. 2004 . “ Empirical processes based on pseudo-observations II: the multivariate case, in Asymptotic methods in stochastics: Festschrift for Miklós Csörgö, ” . Edited by: Horváth , L. and Szyszkowicz , B. Vol. 44 , 381 – 406 . Providence , RI : The Fields Institute Communications Series, American Mathematical Society .
  • Fermanian , J.-D. and Scaillet , O. 2003 . Nonparametric estimation of copulas for time series . J. Risk , 5 : 25 – 54 .
  • Bücher , A. and Dette , H. 2010 . A note on bootstrap approximations for the empirical copula process . Statist. Probab. Lett. , 80 ( 23–24 ) : 1925 – 1932 . (doi:10.1016/j.spl.2010.08.021)
  • Omelka , M. , Gijbels , I. and Veraverbeke , N. 2009 . Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing . Ann. Statist. , 37 ( 5B ) : 3023 – 3058 . (doi:10.1214/08-AOS666)
  • Chen , S. X. and Huang , T.-M. 2007 . Nonparametric estimation of copula functions for dependence modelling . Canad. J. Statist. , 35 ( 2 ) : 265 – 282 . (doi:10.1002/cjs.5550350205)
  • Charpentier , A. , Fermanian , J.-D. and Scaillet , O. 2007 . “ The estimation of copulas: Theory and practice, in Copulas: From Theory to Application in Finance ” . Edited by: Rank , J. 35 – 60 . London : Risk Books .
  • Ruymgaart , F. H. , Shorack , G. R. and van Zwet , W. R. 1972 . Asymptotic normality of nonparametric tests for independence . Ann. Math. Statist. , 43 : 1122 – 1135 . (doi:10.1214/aoms/1177692465)
  • Chung , K.-L. 1949 . An estimate concerning the Kolmogoroff limit distribution . Trans. Amer. Math. Soc. , 67 : 36 – 50 .
  • Stute , W. 1982 . The oscillation behavior of empirical processes . Ann. Probab. , 10 : 86 – 107 . (doi:10.1214/aop/1176993915)
  • Csörgő , M. and Horváth , L. 1988 . A note on strong approximations of multivariate empirical processes . Stochastic Process. Appl. , 28 ( 1 ) : 101 – 109 . (doi:10.1016/0304-4149(88)90068-3)

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