• Bernoulli
  • Volume 23, Number 1 (2017), 743-772.

Weak convergence of the empirical copula process with respect to weighted metrics

Betina Berghaus, Axel Bücher, and Stanislav Volgushev

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The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.

Article information

Bernoulli, Volume 23, Number 1 (2017), 743-772.

Received: November 2014
Revised: June 2015
First available in Project Euclid: 27 September 2016

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bivariate rank statistics empirical copula process Pickands dependence function strongly mixing weighted weak convergence


Berghaus, Betina; Bücher, Axel; Volgushev, Stanislav. Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli 23 (2017), no. 1, 743--772. doi:10.3150/15-BEJ751.

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  • [1] Aistleitner, C. and Dick, J. (2015). Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality. Acta Arith. 167 143–171.
  • [2] Berghaus, B., Bücher, A. and Dette, H. (2013). Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence. J. SFdS 154 116–137.
  • [3] Berghaus, B., Bücher, A. and Volgushev, S. (2015). Supplement to “Weak convergence of the empirical copula process with respect to weighted metrics.” DOI:10.3150/15-BEJ751SUPP.
  • [4] Beutner, E. and Zähle, H. (2010). A modified functional delta method and its application to the estimation of risk functionals. J. Multivariate Anal. 101 2452–2463.
  • [5] Bücher, A., Dette, H. and Volgushev, S. (2011). New estimators of the Pickands dependence function and a test for extreme-value dependence. Ann. Statist. 39 1963–2006.
  • [6] Bücher, A., Segers, J. and Volgushev, S. (2014). When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs. Ann. Statist. 42 1598–1634.
  • [7] Bücher, A. and Volgushev, S. (2013). Empirical and sequential empirical copula processes under serial dependence. J. Multivariate Anal. 119 61–70.
  • [8] Capéraà, P., Fougères, A.-L. and Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika 84 567–577.
  • [9] Chen, X. and Fan, Y. (2006). Estimation of copula-based semiparametric time series models. J. Econometrics 130 307–335.
  • [10] Csörgő, M., Csörgő, S., Horváth, L. and Mason, D.M. (1986). Weighted empirical and quantile processes. Ann. Probab. 14 31–85.
  • [11] Csörgő, M. and Yu, H. (1996). Weak approximations for quantile processes of stationary sequences. Canad. J. Statist. 24 403–430.
  • [12] Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statist. Probab. Lett. 12 429–439.
  • [13] Dudley, R.M. (1992). Fréchet differentiability, $p$-variation and uniform Donsker classes. Ann. Probab. 20 1968–1982.
  • [14] Dudley, R.M. and Norvaiša, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation. Lecture Notes in Math. 1703. Berlin: Springer.
  • [15] Einmahl, J.H.J. (1987). Multivariate Empirical Processes. CWI Tract 32. Amsterdam: Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica.
  • [16] Fermanian, J.-D., Radulović, D. and Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli 10 847–860.
  • [17] Gaenssler, P. and Stute, W. (1987). Seminar on Empirical Processes. DMV Seminar 9. Basel: Birkhäuser.
  • [18] 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.
  • [19] 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.
  • [20] Genest, C. and Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. Ann. Statist. 37 2990–3022.
  • [21] Genest, C. and Segers, J. (2010). On the covariance of the asymptotic empirical copula process. J. Multivariate Anal. 101 1837–1845.
  • [22] Gill, R.D. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method. I. Scand. J. Stat. 16 97–128.
  • [23] Gudendorf, G. and Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. J. Statist. Plann. Inference 142 3073–3085.
  • [24] Hallin, M., Ingenbleek, J.-F. and Puri, M.L. (1985). Linear serial rank tests for randomness against ARMA alternatives. Ann. Statist. 13 1156–1181.
  • [25] Hallin, M. and Puri, M.L. (1988). Optimal rank-based procedures for time series analysis: Testing an ARMA model against other ARMA models. Ann. Statist. 16 402–432.
  • [26] Kley, T., Volgushev, S., Dette, H. and Hallin, M. (2016). Quantile spectral processes: Asymptotic analysis and inference. Bernoulli 22 1770–1807.
  • [27] Owen, A.B. (2005). Multidimensional variation for quasi-Monte Carlo. In Contemporary Multivariate Analysis and Design of Experiments. Ser. Biostat. 2 49–74. Hackensack, NJ: World Sci. Publ.
  • [28] Pickands, J. III (1981). Multivariate extreme value distributions. In Proceedings of the 43rd Session of the International Statistical Institute, Vol. 2 (Buenos Aires, 1981) 49 859–878, 894–902. International Statistical Institute.
  • [29] Rio, E. (2000). Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Mathématiques & Applications (Berlin) [Mathematics & Applications] 31. Berlin: Springer.
  • [30] Rojo Jiménez, J., Villa-Diharce, E. and Flores, M. (2001). Nonparametric estimation of the dependence function in bivariate extreme value distributions. J. Multivariate Anal. 76 159–191.
  • [31] Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 4 912–923.
  • [32] Segers, J. (2012). Asymptotics of empirical copula processes under non-restrictive smoothness assumptions. Bernoulli 18 764–782.
  • [33] Shao, Q.-M. and Yu, H. (1996). Weak convergence for weighted empirical processes of dependent sequences. Ann. Probab. 24 2098–2127.
  • [34] 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.
  • [35] Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. Ann. Probab. 12 361–379.
  • [36] Tsukahara, H. (2005). Semiparametric estimation in copula models. Canad. J. Statist. 33 357–375.
  • [37] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes. Springer Series in Statistics. New York: Springer.
  • [38] van der Vaart, A.W. and Wellner, J.A. (2007). Empirical processes indexed by estimated functions. In Asymptotics: Particles, Processes and Inverse Problems. Institute of Mathematical Statistics Lecture Notes—Monograph Series 55 234–252. Beachwood, OH: IMS.
  • [39] Ziemer, W.P. (1989). Weakly Differentiable Functions. Graduate Texts in Mathematics 120. New York: Springer.

Supplemental materials

  • Supplement to “Weak convergence of the empirical copula process with respect to weighted metrics”. A detailed exposition on bounded variation and Lebesgue–Stieltjes integration for two-variate functions and the proofs of Proposition 4.4 and of Lemma 4.9 can be found in [3].