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Semiparametric Multivariate Density Estimation for Positive Data Using Copulas

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  • Taoufik Bouezmarni
  • Jeroen V.K. Rombouts

    ()
    (IEA, HEC Montréal)

Abstract

In this paper we estimate density functions for positive multivariate data. We propose a semiparametric approach. The estimator combines gamma kernels or local linear kernels, also called boundary kernels, for the estimation of the marginal densities with semiparametric copulas to model the dependence. This semiparametric approach is robust both to the well known boundary bias problem and the curse of dimensionality problem. We derive the mean integrated squared error properties, including the rate of convergence, the uniform strong consistency and the asymptotic normality. A simulation study investigates the finite sample performance of the estimator. We find that univariate least squares cross validation, to choose the bandwidth for the estimation of the marginal densities, works well and that the estimator we propose performs very well also for data with unbounded support. Applications in the field of finance are provided.

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Bibliographic Info

Paper provided by HEC Montréal, Institut d'économie appliquée in its series Cahiers de recherche with number 07-08.

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Length: 32 pages
Date of creation: Jul 2007
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Handle: RePEc:iea:carech:0708

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Keywords: Asymptotic properties; asymmetric kernels; boundary bias; copula; curse of dimension; least squares cross validation.;

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References

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  1. Gustavo Grullon & Roni Michaely, 2002. "Dividends, Share Repurchases, and the Substitution Hypothesis," Journal of Finance, American Finance Association, American Finance Association, vol. 57(4), pages 1649-1684, 08.
  2. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 20(03), pages 535-562, June.
  3. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer, Springer, vol. 52(3), pages 471-480, September.
  4. Taoufik Bouezmarni & Jeroen V.K. Rombouts, 2006. "Nonparametric Density Estimation for Positive Time Series," Cahiers de recherche, HEC Montréal, Institut d'économie appliquée 06-09, HEC Montréal, Institut d'économie appliquée.
  5. Bouezmarni, Taoufik & Scaillet, Olivier, 2005. "Consistency Of Asymmetric Kernel Density Estimators And Smoothed Histograms With Application To Income Data," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 21(02), pages 390-412, April.
  6. Chen, Xiaohong & Fan, Yanqin & Tsyrennikov, Viktor, 2006. "Efficient Estimation of Semiparametric Multivariate Copula Models," Journal of the American Statistical Association, American Statistical Association, American Statistical Association, vol. 101, pages 1228-1240, September.
  7. Cho, Myeong-Hyeon, 1998. "Ownership structure, investment, and the corporate value: an empirical analysis," Journal of Financial Economics, Elsevier, Elsevier, vol. 47(1), pages 103-121, January.
  8. Kim, Gunky & Silvapulle, Mervyn J. & Silvapulle, Paramsothy, 2007. "Comparison of semiparametric and parametric methods for estimating copulas," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 51(6), pages 2836-2850, March.
  9. Joe, Harry, 2005. "Asymptotic efficiency of the two-stage estimation method for copula-based models," Journal of Multivariate Analysis, Elsevier, Elsevier, vol. 94(2), pages 401-419, June.
  10. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation of copula-based semiparametric time series models," Journal of Econometrics, Elsevier, Elsevier, vol. 130(2), pages 307-335, February.
  11. Lejeune, Michel & Sarda, Pascal, 1992. "Smooth estimators of distribution and density functions," Computational Statistics & Data Analysis, Elsevier, Elsevier, vol. 14(4), pages 457-471, November.
  12. H. G. Müller & U. Stadtmüller, 1999. "Multivariate boundary kernels and a continuous least squares principle," Journal of the Royal Statistical Society Series B, Royal Statistical Society, Royal Statistical Society, vol. 61(2), pages 439-458.
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Cited by:
  1. Diers, Dorothea & Eling, Martin & Marek, Sebastian D., 2012. "Dependence modeling in non-life insurance using the Bernstein copula," Insurance: Mathematics and Economics, Elsevier, vol. 50(3), pages 430-436.
  2. Zhao, XiaoBing & Zhou, Xian, 2010. "Applying copula models to individual claim loss reserving methods," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 290-299, April.

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