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Approximation of Multidimensional Stable Densities

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  • Byczkowski, T.
  • Nolan, J. P.
  • Rajput, B.

Abstract

Multivariate stable densities do not generally have explicit formula, but they can be specified indirectly by means of a spectral measure. Our main result gives an approximation that is used for numerical computation of these densities. We construct a discrete spectral measure, with explicit formulas for the number, location, and magnitudes of the atoms, that yields a stable density uniformly close to the original one. Sample graphs of two dimensional stable densities with dependence are given.

Suggested Citation

  • Byczkowski, T. & Nolan, J. P. & Rajput, B., 1993. "Approximation of Multidimensional Stable Densities," Journal of Multivariate Analysis, Elsevier, vol. 46(1), pages 13-31, July.
  • Handle: RePEc:eee:jmvana:v:46:y:1993:i:1:p:13-31
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    Cited by:

    1. Abdul-Hamid, Husein & Nolan, John P., 1998. "Multivariate Stable Densities as Functions of One Dimensional Projections," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 80-89, October.
    2. Mohammad Mohammadi & Adel Mohammadpour & Hiroaki Ogata, 2015. "On estimating the tail index and the spectral measure of multivariate $$\alpha $$ α -stable distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 78(5), pages 549-561, July.
    3. Davydov, Yu. & Nagaev, A. V., 2002. "On Two Aproaches to Approximation of Multidimensional Stable Laws," Journal of Multivariate Analysis, Elsevier, vol. 82(1), pages 210-239, July.
    4. Peters, G.W. & Sisson, S.A. & Fan, Y., 2012. "Likelihood-free Bayesian inference for α-stable models," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3743-3756.
    5. repec:ere:journl:v:xxxvii:y:2018:i:1:p:43-76 is not listed on IDEAS
    6. John Nolan, 2013. "Multivariate elliptically contoured stable distributions: theory and estimation," Computational Statistics, Springer, vol. 28(5), pages 2067-2089, October.
    7. Matsui, Muneya & Takemura, Akimichi, 2009. "Integral representations of one-dimensional projections for multivariate stable densities," Journal of Multivariate Analysis, Elsevier, vol. 100(3), pages 334-344, March.
    8. Ogata, Hiroaki, 2013. "Estimation for multivariate stable distributions with generalized empirical likelihood," Journal of Econometrics, Elsevier, vol. 172(2), pages 248-254.

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