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Estimating copula densities, using model selection techniques

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  • Kallenberg, Wilbert C.M.

Abstract

Recently a new way of modeling dependence has been introduced considering a sequence of parametric copula models, covering more and more dependency aspects and thus giving a closer approximation to the true copula density. The method uses contamination families based on Legendre polynomials. It has been shown that in general after a few steps accurate approximations are obtained. In this paper selection of the adequate number of steps is considered, and estimation of the unknown parameters within the chosen contamination family is established, thus obtaining an estimator of the unknown copula density. There should be a balance between the complexity of the model and the number of parameters to be estimated. High complexity gives a low model error, but a large stochastic or estimation error, while a very simple model gives a small stochastic error, but a large model error. Techniques from model selection are applied, thus letting the data tell us which aspects are important enough to capture into the model. Natural and simple estimators of the involved Fourier coefficients complete the procedure. Theoretical results show that the expected quadratic error is reduced by the selection rule to the same order of magnitude as in a classical parametric problem. The method is applied on a real data set, illustrating that the new method describes the data set very well: the error involved in the classical Gaussian copula density is reduced with no fewer than 50%.

Suggested Citation

  • Kallenberg, Wilbert C.M., 2009. "Estimating copula densities, using model selection techniques," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 209-223, October.
  • Handle: RePEc:eee:insuma:v:45:y:2009:i:2:p:209-223
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    References listed on IDEAS

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    1. Genest, Christian & Rémillard, Bruno & Beaudoin, David, 2009. "Goodness-of-fit tests for copulas: A review and a power study," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 199-213, April.
    2. Panchenko, Valentyn, 2005. "Goodness-of-fit test for copulas," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 355(1), pages 176-182.
    3. Klugman, Stuart A. & Parsa, Rahul, 1999. "Fitting bivariate loss distributions with copulas," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 139-148, March.
    4. Fermanian, Jean-David, 2005. "Goodness-of-fit tests for copulas," Journal of Multivariate Analysis, Elsevier, vol. 95(1), pages 119-152, July.
    5. Biau, Gérard & Wegkamp, Marten, 2005. "A note on minimum distance estimation of copula densities," Statistics & Probability Letters, Elsevier, vol. 73(2), pages 105-114, June.
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    Cited by:

    1. Ledwina, Teresa & Wyłupek, Grzegorz, 2014. "Validation of positive quadrant dependence," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 38-47.

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