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Finite Gaussian Mixture Approximations to Analytically Intractable Density Kernels

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  • Khorunzhina, Natalia
  • Richard, Jean-Francois

Abstract

The objective of the paper is that of constructing finite Gaussian mixture approximations to analytically intractable density kernels. The proposed method is adaptive in that terms are added one at the time and the mixture is fully re-optimized at each step using a distance measure that approximates the corresponding importance sampling variance. All functions of interest are evaluated under Gaussian quadrature rules. Examples include a sequential (filtering) evaluation of the likelihood function of a stochastic volatility model where all relevant densities (filtering, predictive and likelihood) are closely approximated by mixtures.

Suggested Citation

  • Khorunzhina, Natalia & Richard, Jean-Francois, 2016. "Finite Gaussian Mixture Approximations to Analytically Intractable Density Kernels," MPRA Paper 72326, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:72326
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    1. Roman Liesenfeld & Jean-Francois Richard, 2006. "Classical and Bayesian Analysis of Univariate and Multivariate Stochastic Volatility Models," Econometric Reviews, Taylor & Francis Journals, vol. 25(2-3), pages 335-360.
    2. Heiss, Florian & Winschel, Viktor, 2008. "Likelihood approximation by numerical integration on sparse grids," Journal of Econometrics, Elsevier, vol. 144(1), pages 62-80, May.
    3. Geweke, John, 1996. "Monte carlo simulation and numerical integration," Handbook of Computational Economics, in: H. M. Amman & D. A. Kendrick & J. Rust (ed.), Handbook of Computational Economics, edition 1, volume 1, chapter 15, pages 731-800, Elsevier.
    4. Tucker, Alan L, 1992. "A Reexamination of Finite- and Infinite-Variance Distributions as Models of Daily Stock Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 10(1), pages 73-81, January.
    5. Keane, Michael P & Wolpin, Kenneth I, 1997. "The Career Decisions of Young Men," Journal of Political Economy, University of Chicago Press, vol. 105(3), pages 473-522, June.
    6. Hiroyuki Kasahara & Katsumi Shimotsu, 2015. "Testing the Number of Components in Normal Mixture Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(512), pages 1632-1645, December.
    7. Stephen V. Cameron & James J. Heckman, 2001. "The Dynamics of Educational Attainment for Black, Hispanic, and White Males," Journal of Political Economy, University of Chicago Press, vol. 109(3), pages 455-499, June.
    8. Ardia, David & Hoogerheide, Lennart F. & van Dijk, Herman K., 2009. "Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 29(i03).
    9. Hoogerheide, Lennart F. & Kaashoek, Johan F. & van Dijk, Herman K., 2007. "On the shape of posterior densities and credible sets in instrumental variable regression models with reduced rank: An application of flexible sampling methods using neural networks," Journal of Econometrics, Elsevier, vol. 139(1), pages 154-180, July.
    10. Ferreira, Jose T.A.S. & Steel, Mark F.J., 2007. "Model comparison of coordinate-free multivariate skewed distributions with an application to stochastic frontiers," Journal of Econometrics, Elsevier, vol. 137(2), pages 641-673, April.
    11. Wayne S. DeSarbo & Alexandru M. Degeratu & Michel Wedel & M. Kim Saxton, 2001. "The Spatial Representation of Market Information," Marketing Science, INFORMS, vol. 20(4), pages 426-441, June.
    12. James H. Stock & Mark W. Watson, 2007. "Why Has U.S. Inflation Become Harder to Forecast?," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 39(s1), pages 3-33, February.
    13. Hoogerheide, Lennart & Opschoor, Anne & van Dijk, Herman K., 2012. "A class of adaptive importance sampling weighted EM algorithms for efficient and robust posterior and predictive simulation," Journal of Econometrics, Elsevier, vol. 171(2), pages 101-120.
    14. Stefano Mazzuco & Bruno Scarpa, 2015. "Fitting age-specific fertility rates by a flexible generalized skew normal probability density function," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 178(1), pages 187-203, January.
    15. Kon, Stanley J, 1984. "Models of Stock Returns-A Comparison," Journal of Finance, American Finance Association, vol. 39(1), pages 147-165, March.
    16. repec:dau:papers:123456789/6072 is not listed on IDEAS
    17. Sangjoon Kim & Neil Shephard & Siddhartha Chib, 1998. "Stochastic Volatility: Likelihood Inference and Comparison with ARCH Models," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 361-393.
    18. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    19. Subu Venkataraman, 1997. "Value at risk for a mixture of normal distributions: the use of quasi- Bayesian estimation techniques," Economic Perspectives, Federal Reserve Bank of Chicago, vol. 21(Mar), pages 2-13.
    20. H. M. Amman & D. A. Kendrick & J. Rust (ed.), 1996. "Handbook of Computational Economics," Handbook of Computational Economics, Elsevier, edition 1, volume 1, number 1.
    21. Ardia, David & Hoogerheide, Lennart F. & van Dijk, Herman K., 2009. "Adaptive Mixture of Student-t Distributions as a Flexible Candidate Distribution for Efficient Simulation: The R Package AdMit," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 29(i03).
    22. Yulia V. Marchenko & Marc G. Genton, 2012. "A Heckman Selection- t Model," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(497), pages 304-317, March.
    23. Emmanuel O. Ogundimu & Jane L. Hutton, 2016. "A Sample Selection Model with Skew-normal Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(1), pages 172-190, March.
    24. Jean-Francois Richard, 2007. "Efficient High-Dimensional Importance Sampling," Working Paper 321, Department of Economics, University of Pittsburgh, revised Jan 2007.
    25. Omori, Yasuhiro & Chib, Siddhartha & Shephard, Neil & Nakajima, Jouchi, 2007. "Stochastic volatility with leverage: Fast and efficient likelihood inference," Journal of Econometrics, Elsevier, vol. 140(2), pages 425-449, October.
    26. C. Adcock, 2010. "Asset pricing and portfolio selection based on the multivariate extended skew-Student-t distribution," Annals of Operations Research, Springer, vol. 176(1), pages 221-234, April.
    27. Richard, Jean-Francois & Zhang, Wei, 2007. "Efficient high-dimensional importance sampling," Journal of Econometrics, Elsevier, vol. 141(2), pages 1385-1411, December.
    28. Duong, Tarn, 2007. "ks: Kernel Density Estimation and Kernel Discriminant Analysis for Multivariate Data in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 21(i07).
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    More about this item

    Keywords

    Finite mixture; Distance measure; Gaussian quadrature; Importance sampling; Adaptive algorithm; Stochastic volatility; Density kernel;
    All these keywords.

    JEL classification:

    • C11 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Bayesian Analysis: General
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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