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The Class of ( p , q )-spherical Distributions with an Extension of the Sector and Circle Number Functions

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  • Wolf-Dieter Richter

    (University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany)

Abstract

For evaluating the probabilities of arbitrary random events with respect to a given multivariate probability distribution, specific techniques are of great interest. An important two-dimensional high risk limit law is the Gauss-exponential distribution whose probabilities can be dealt with based on the Gauss–Laplace law. The latter will be considered here as an element of the newly-introduced family of ( p , q ) -spherical distributions. Based on a suitably-defined non-Euclidean arc-length measure on ( p , q ) -circles, we prove geometric and stochastic representations of these distributions and correspondingly distributed random vectors, respectively. These representations allow dealing with the new probability measures similarly to with elliptically-contoured distributions and more general homogeneous star-shaped ones. This is demonstrated by the generalization of the Box–Muller simulation method. In passing, we prove an extension of the sector and circle number functions.

Suggested Citation

  • Wolf-Dieter Richter, 2017. "The Class of ( p , q )-spherical Distributions with an Extension of the Sector and Circle Number Functions," Risks, MDPI, vol. 5(3), pages 1-17, July.
  • Handle: RePEc:gam:jrisks:v:5:y:2017:i:3:p:40-:d:105540
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    References listed on IDEAS

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    1. Henschel, V. & Richter, W. -D., 2002. "Geometric Generalization of the Exponential Law," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 189-204, May.
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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.
    2. Liebscher Eckhard & Richter Wolf-Dieter, 2020. "Modelling with star-shaped distributions," Dependence Modeling, De Gruyter, vol. 8(1), pages 45-69, January.
    3. Liebscher Eckhard & Richter Wolf-Dieter, 2020. "Modelling with star-shaped distributions," Dependence Modeling, De Gruyter, vol. 8(1), pages 45-69, January.
    4. Wolf-Dieter Richter & Vincent Wenzel, 2019. "Mysterious Circle Numbers. Does π p,q Approach π p When q Is Tending to p ?," Mathematics, MDPI, vol. 7(9), pages 1-8, September.
    5. Wolf-Dieter Richter, 2019. "On Ball Numbers," Mathematics, MDPI, vol. 7(8), pages 1-10, August.

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