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On p-generalized elliptical random processes

Author

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  • Klaus Müller

    (University of Rostock, Institute of Mathematics)

  • Wolf-Dieter Richter

    (University of Rostock, Institute of Mathematics)

Abstract

We introduce rank-k-continuous axis-aligned p-generalized elliptically contoured distributions and study their properties such as stochastic representations, moments, and density-like representations. Applying the Kolmogorov existence theorem, we prove the existence of random processes having axis-aligned p-generalized elliptically contoured finite dimensional distributions with arbitrary location and scale functions and a consistent sequence of density generators of p-generalized spherical invariant distributions. Particularly, we consider scale mixtures of rank-k-continuous axis-aligned p-generalized elliptically contoured Gaussian distributions and answer the question when an n-dimensional rank-k-continuous axis-aligned p-generalized elliptically contoured distribution is representable as a scale mixture of n-dimensional rank-k-continuous p-generalized Gaussian distribution for a suitable mixture distribution of a positive random variable. Based on this class of multivariate probability distributions, we introduce scale mixed p-generalized Gaussian processes having axis-aligned finite dimensional distributions being p-generalizations of elliptical random processes. Additionally, some of their characteristic properties are discussed and approximates of trajectories of several examples such as p-generalized Student-t and p-generalized Slash processes having axis-aligned finite dimensional distributions are simulated with the help of algorithms to simulate rank-k-continuous axis-aligned p-generalized elliptically contoured distributions.

Suggested Citation

  • Klaus Müller & Wolf-Dieter Richter, 2019. "On p-generalized elliptical random processes," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-37, December.
    • Handle: RePEc:spr:jstada:v:6:y:2019:i:1:d:10.1186_s40488-019-0090-6
    DOI: 10.1186/s40488-019-0090-6
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    References listed on IDEAS

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    1. Choy, S.T. Boris & Chan, C.M., 2003. "Scale Mixtures Distributions in Insurance Applications," ASTIN Bulletin, Cambridge University Press, vol. 33(1), pages 93-104, May.
    2. Wolf-Dieter Richter, 2016. "Exact inference on scaling parameters in norm and antinorm contoured sample distributions," Journal of Statistical Distributions and Applications, Springer, vol. 3(1), pages 1-16, December.
    3. Wolf-Dieter Richter, 2017. "Statistical reasoning in dependent p-generalized elliptically contoured distributions and beyond," Journal of Statistical Distributions and Applications, Springer, vol. 4(1), pages 1-25, December.
    4. Reinaldo B. Arellano‐Valle & Adelchi Azzalini, 2006. "On the Unification of Families of Skew‐normal Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(3), pages 561-574, September.
    5. Kano, Y., 1994. "Consistency Property of Elliptic Probability Density Functions," Journal of Multivariate Analysis, Elsevier, vol. 51(1), pages 139-147, October.
    6. Müller K. & Richter W.-D., 2016. "Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables," Dependence Modeling, De Gruyter, vol. 4(1), pages 1-33, February.
    7. Huang, Steel T. & Cambanis, Stamatis, 1979. "Spherically invariant processes: Their nonlinear structure, discrimination, and estimation," Journal of Multivariate Analysis, Elsevier, vol. 9(1), pages 59-83, March.
    8. Klaus Müller & Wolf-Dieter Richter, 2015. "Exact extreme value, product, and ratio distributions under non-standard assumptions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 99(1), pages 1-30, January.
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    Cited by:

    1. Wolf-Dieter Richter, 2019. "On (p1,…,pk)-spherical distributions," Journal of Statistical Distributions and Applications, Springer, vol. 6(1), pages 1-18, December.

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