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GeD spline estimation of multivariate Archimedean copulas

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Author Info

  • Dimitrova, Dimitrina S.
  • Kaishev, Vladimir K.
  • Penev, Spiridon I.

Abstract

A new multivariate Archimedean copula estimation method is proposed in a non-parametric setting. The method uses the so-called Geometrically Designed splines (GeD splines) to represent the cdf of a random variable W[theta], obtained through the probability integral transform of an Archimedean copula with parameter [theta]. Sufficient conditions for the GeD spline estimator to possess the properties of the underlying theoretical cdf, K([theta],t), of W[theta], are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem are separated. Thus, the resulting spline estimator is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is truly multivariate, it brings about numerical efficiency and as a result can be applied with large volumes of data and for dimensions d>=2, as illustrated by the numerical examples presented.

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Bibliographic Info

Article provided by Elsevier in its journal Computational Statistics & Data Analysis.

Volume (Year): 52 (2008)
Issue (Month): 7 (March)
Pages: 3570-3582

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Handle: RePEc:eee:csdana:v:52:y:2008:i:7:p:3570-3582

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  1. Joe, Harry, 1990. "Multivariate concordance," Journal of Multivariate Analysis, Elsevier, vol. 35(1), pages 12-30, October.
  2. Chen, Xiaohong & Fan, Yanqin & Tsyrennikov, Viktor, 2006. "Efficient Estimation of Semiparametric Multivariate Copula Models," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1228-1240, September.
  3. Peter Hall & Natalie Neumeyer, 2006. "Estimating a bivariate density when there are extra data on one or both components," Biometrika, Biometrika Trust, vol. 93(2), pages 439-450, June.
  4. Christian Genest & Jean-François Quessy & Bruno Rémillard, 2006. "Goodness-of-fit Procedures for Copula Models Based on the Probability Integral Transformation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics & Finnish Statistical Society & Norwegian Statistical Association & Swedish Statistical Association, vol. 33(2), pages 337-366.
  5. Lambert, Philippe, 2007. "Archimedean copula estimation using Bayesian splines smoothing techniques," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6307-6320, August.
  6. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(03), pages 535-562, June.
  7. Genest, Christian & Rivest, Louis-Paul, 2001. "On the multivariate probability integral transformation," Statistics & Probability Letters, Elsevier, vol. 53(4), pages 391-399, July.
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Cited by:
  1. Diks, Cees & Panchenko, Valentyn & van Dijk, Dick, 2010. "Out-of-sample comparison of copula specifications in multivariate density forecasts," Journal of Economic Dynamics and Control, Elsevier, vol. 34(9), pages 1596-1609, September.
  2. Hernández-Lobato, José Miguel & Suárez, Alberto, 2011. "Semiparametric bivariate Archimedean copulas," Computational Statistics & Data Analysis, Elsevier, vol. 55(6), pages 2038-2058, June.
  3. Christian Genest & Johanna Nešlehová & Johanna Ziegel, 2011. "Inference in multivariate Archimedean copula models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer, vol. 20(2), pages 223-256, August.

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