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Multivariate Stable Densities as Functions of One Dimensional Projections

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  • Abdul-Hamid, Husein
  • Nolan, John P.

Abstract

The density of a generald-dimensional stable random vectorXis expressed as an integral over the sphere in dof a function of the parameters of the one dimensional projections ofX. These formulas give insight into the form of multivariate stable densities and are useful for numerical calculations. Corollaries give simplified expressions for symmetric stable and the[alpha]=1 strictly stable densities, relations among the densities in different dimensions, and values of the densities at the location parameter for all cases except the[alpha]=1, non-strictly stable ones. Expressions for the densities in the multidimensional analog of Zolotarev's (M) parameterization and a discussion of computational versions of the formulas are also given.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:jmvana:v:67:y:1998:i:1:p:80-89
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    References listed on IDEAS

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    1. 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.
    2. Nolan, John P., 1998. "Parameterizations and modes of stable distributions," Statistics & Probability Letters, Elsevier, vol. 38(2), pages 187-195, June.
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    4. Tsionas, Mike G., 2016. "Bayesian analysis of multivariate stable distributions using one-dimensional projections," Journal of Multivariate Analysis, Elsevier, vol. 143(C), pages 185-193.
    5. John Nolan, 2013. "Multivariate elliptically contoured stable distributions: theory and estimation," Computational Statistics, Springer, vol. 28(5), pages 2067-2089, October.
    6. Yousef, Waleed A. & Kundu, Subrata, 2014. "Learning algorithms may perform worse with increasing training set size: Algorithm–data incompatibility," Computational Statistics & Data Analysis, Elsevier, vol. 74(C), pages 181-197.
    7. Nolan, John P., 2018. "Truncated fractional moments of stable laws," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 312-318.
    8. 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.
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