Bernoulli

  • Bernoulli
  • Volume 10, Number 5 (2004), 847-860.

Weak convergence of empirical copula processes

Jean-David Fermanian, Dragan Radulovic, and Marten Wegkamp

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Abstract

Weak convergence of the empirical copula process has been established by Deheuvels in the case of independent marginal distributions. Van der Vaart and Wellner utilize the functional delta method to show convergence in $\ell^\infty([a,b]^2)$> for some 0<a<b<1, under restrictions on the distribution functions. We extend their results by proving the weak convergence of this process in $\ell^\infty([0,1]^2)$> under minimal conditions on the copula function, which coincides with the result obtained by Gaenssler and Stute. It is argued that the condition on the copula function is necessary. The proof uses the functional delta method and, as a consequence, the convergence of the bootstrap counterpart of the empirical copula process follows immediately. In addition, weak convergence of the smoothed empirical copula process is established.

Article information

Source
Bernoulli Volume 10, Number 5 (2004), 847-860.

Dates
First available in Project Euclid: 4 November 2004

Permanent link to this document
http://projecteuclid.org/euclid.bj/1099579158

Digital Object Identifier
doi:10.3150/bj/1099579158

Mathematical Reviews number (MathSciNet)
MR2093613

Zentralblatt MATH identifier
02149083

Keywords
empirical copula process smoothed empirical copula processes weak convergence

Citation

Fermanian, Jean-David; Radulovic, Dragan; Wegkamp, Marten. Weak convergence of empirical copula processes. Bernoulli 10 (2004), no. 5, 847--860. doi:10.3150/bj/1099579158. http://projecteuclid.org/euclid.bj/1099579158.


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References

  • [1] Bouyé, E., Durrleman, V., Nikeghbali, A., Riboulet, G. and Roncalli, T. (2000) Copulas for Finance. A Reading Guide and Some Applications. Paris: Groupe de Recherche Opérationnelle, Crédit Lyonnais.
  • [2] Capéraà, P., Fougères, A.-L. and Genest, C. (1997) A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567-577.
  • [3] Deheuvels, P. (1979) La fonction de dépendance empirique et ses propriétés. Acad. Roy. Belg. Bull. Cl. Sci. (5), 65, 274-292.
  • [4] Deheuvels, P. (1981a) A Kolmogorov-Smirnov type test for independence and multivariate samples. Rev. Roumaine Math. Pures Appl., 26(2) 213-226.
  • [5] Deheuvels, P. (1981b) A nonparametric test for independence. Publ. Inst. Statist. Univ. Paris., 26(2), 29-50.
  • [6] Embrechts, P., McNeil, A. and Straumann, D. (2002) Correlation and dependence in risk management: properties and pitfalls. In M.A.H. Dempster (ed.), Risk Management: Value at Risk and Beyond, pp. 176-223. Cambridge: Cambridge University Press.
  • [7] Fermanian, J-D. (1996) Multivariate hazard rates under random censorship. CREST Working Paper 9603. Abstract can also be found in the ISI/STMA publication
  • [8] Gaenssler, P. and Stute, W. (1987) Seminar on Empirical Processes, DMV Sem. 9. Basel: Birkhäuser.
  • [9] Genest, C. and MacKay, R.J. (1986a) Copules archimédiennes et familles de lois bidimensionnelles dont les marges sont données. Rev. Canad. Statist., 14, 145-159.
  • [10] Genest, C. and MacKay, R.J. (1986b) The joy of copulas: Bivariate distributions with uniform marginals. Amer. Statist., 40, 280-283.
  • [11] Genest, C. and Rivest, L.-P. (1993) Statistical inference procedures for bivariate Archimedean copulas. J. Amer. Statist. Assoc., 88, 1034-1043. Abstract can also be found in the ISI/STMA publication
  • [12] 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. Abstract can also be found in the ISI/STMA publication
  • [13] Joe, H. (1997) Multivariate Models and Dependence Concepts. London: Chapman and Hall.
  • [14] Nelsen, R.B. (1999) An Introduction to Copulas, Lecture Notes in Statist. 139. New York: Springer- Verlag.
  • [15] Neuhaus, G. (1971) On the weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist., 42, 1285-1295.
  • [16] Rüschendorf, L. (1976) Asymptotic distributions of multivariate rank order statistics. Ann. Statist., 4, 912-923.
  • [17] Ruymgaart, F.H. (1973) Asymptotic Theory for Rank Tests for Independence, MC Tract 43. Amsterdam: Mathematisch Instituut.
  • [18] Ruymgaart, F.H. (1974) Asymptotic normality of nonparametric tests for independence. Ann. Statist., 2, 892-910.
  • [19] 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.
  • [20] Schönbucher, P. and Schubert, D. (2000) Copula dependent default risk in intensity models. Preprint, Department of Statistics, Bonn University.
  • [21] Schweizer, B. (1991) Thirty years of copulas. In G. Dall´Aglio, S. Kotz and G. Salinetti (eds), Advances in Probability Distributions with Given Marginals. Dordrecht: Kluwer.
  • [22] Schweizer, B. and Sklar, A. (1974) Operations on distribution functions not derivable from operations on random variables. Studia Math., 52, 43-52.
  • [23] Sklar, A. (1959) Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris, 8, 229-231.
  • [24] van der Vaart, A.W. (1994) Weak convergence of smoothed empirical processes. Scand. J. Statist., 21, 501-504. Abstract can also be found in the ISI/STMA publication
  • [25] van der Vaart, A.W. and Wellner, J.A. (1996) Weak Convergence and Empirical Processes. New York: Springer-Verlag. Abstract can also be found in the ISI/STMA publication