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Approximation of Multidimensional Stable Densities

Author

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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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    Citations

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    Cited by:

    1. Szabolcs Majoros & Andr'as Zempl'eni, 2018. "Multivariate stable distributions and their applications for modelling cryptocurrency-returns," Papers 1810.09521, arXiv.org.
    2. Joshua Rushton, 2007. "A Functional LIL for d-Dimensional Stable Processes; Invariance for Lévy- and Other Weakly Convergent Processes," Journal of Theoretical Probability, Springer, vol. 20(3), pages 397-427, September.
    3. 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.
    4. 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.
    5. Ramona Serrano Bautista & Leovardo Mata Mata, 2018. "Estimación del VaR mediante un modelo condicional multivariado bajo la hipótesis α-estable sub-Gaussiana. (A conditional approach to VaR with multivariate α-stable sub-Gaussian distributions)," Ensayos Revista de Economia, Universidad Autonoma de Nuevo Leon, Facultad de Economia, vol. 0(1), pages 43-76, May.
    6. John Nolan, 2013. "Multivariate elliptically contoured stable distributions: theory and estimation," Computational Statistics, Springer, vol. 28(5), pages 2067-2089, October.
    7. 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.
    8. 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.
    9. 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.
    10. Ogata, Hiroaki, 2013. "Estimation for multivariate stable distributions with generalized empirical likelihood," Journal of Econometrics, Elsevier, vol. 172(2), pages 248-254.
    11. Michael Grabchak, 2021. "On the transition laws of p-tempered $$\alpha $$ α -stable OU-processes," Computational Statistics, Springer, vol. 36(2), pages 1415-1436, June.
    12. Karling, Maicon J. & Lopes, Sílvia R.C. & de Souza, Roberto M., 2023. "Multivariate α-stable distributions: VAR(1) processes, measures of dependence and their estimations," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    13. Balram S. Rajput & Kavi Rama-Murthy, 2007. "Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications," Journal of Theoretical Probability, Springer, vol. 20(1), pages 87-105, March.
    14. Paola Stolfi & Mauro Bernardi & Lea Petrella, 2018. "The sparse method of simulated quantiles: An application to portfolio optimization," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 72(3), pages 375-398, August.
    15. Dominicy, Yves & Heikkilä, Matias & Ilmonen, Pauliina & Veredas, David, 2020. "Flexible multivariate Hill estimators," Journal of Econometrics, Elsevier, vol. 217(2), pages 398-410.

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