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Semiparametric Multivariate Density Estimation for Positive Data Using Copulas

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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File URL: http://www.hec.ca/iea/cahiers/2007/iea0708_jrombouts.pdf
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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
Date of revision:
Handle: RePEc:iea:carech:0708
Contact details of provider: Postal: Institut d'économie appliquée HEC Montréal 3000, Chemin de la Côte-Sainte-Catherine Montréal, Québec H3T 2A7
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  1. Cho, Myeong-Hyeon, 1998. "Ownership structure, investment, and the corporate value: an empirical analysis," Journal of Financial Economics, Elsevier, vol. 47(1), pages 103-121, January.
  2. Taoufik Bouezmarni & Jeroen V.K. Rombouts, 2006. "Nonparametric Density Estimation for Positive Time Series," Cahiers de recherche 06-09, HEC Montréal, Institut d'économie appliquée.
  3. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation of copula-based semiparametric time series models," Journal of Econometrics, Elsevier, vol. 130(2), pages 307-335, February.
  4. Lejeune, Michel & Sarda, Pascal, 1992. "Smooth estimators of distribution and density functions," Computational Statistics & Data Analysis, Elsevier, vol. 14(4), pages 457-471, November.
  5. Kim, Gunky & Silvapulle, Mervyn J. & Silvapulle, Paramsothy, 2007. "Comparison of semiparametric and parametric methods for estimating copulas," Computational Statistics & Data Analysis, Elsevier, vol. 51(6), pages 2836-2850, March.
  6. Bouezmarni, Taoufik & Scaillet, Olivier, 2005. "Consistency Of Asymmetric Kernel Density Estimators And Smoothed Histograms With Application To Income Data," Econometric Theory, Cambridge University Press, vol. 21(02), pages 390-412, April.
  7. 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.
  8. Song Chen, 2000. "Probability Density Function Estimation Using Gamma Kernels," Annals of the Institute of Statistical Mathematics, Springer, vol. 52(3), pages 471-480, September.
  9. Gustavo Grullon & Roni Michaely, 2002. "Dividends, Share Repurchases, and the Substitution Hypothesis," Journal of Finance, American Finance Association, vol. 57(4), pages 1649-1684, 08.
  10. Xiaohong Chen & Yanqin Fan & Victor Tsyrennifov, 2004. "Efficient Estimation of Semiparametric Multivariate Copula Models," Vanderbilt University Department of Economics Working Papers 0420, Vanderbilt University Department of Economics.
  11. Joe, Harry, 2005. "Asymptotic efficiency of the two-stage estimation method for copula-based models," Journal of Multivariate Analysis, Elsevier, vol. 94(2), pages 401-419, June.
  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, vol. 61(2), pages 439-458.
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