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Risks 2017, 5(3), 40; doi:10.3390/risks5030040

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

University of Rostock, Institute of Mathematics, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany
Academic Editor: Mogens Steffensen
Received: 24 May 2017 / Revised: 17 July 2017 / Accepted: 19 July 2017 / Published: 21 July 2017
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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. View Full-Text
Keywords: Gauss-exponential distribution; Gauss–Laplace distribution; stochastic vector representation; geometric measure representation; (p,q)-generalized polar coordinates; (p,q)-arc length; dynamic intersection proportion function; (p,q)-generalized Box–Muller simulation method; (p,q)-spherical uniform distribution; dynamic geometric disintegration Gauss-exponential distribution; Gauss–Laplace distribution; stochastic vector representation; geometric measure representation; (p,q)-generalized polar coordinates; (p,q)-arc length; dynamic intersection proportion function; (p,q)-generalized Box–Muller simulation method; (p,q)-spherical uniform distribution; dynamic geometric disintegration
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Richter, W.-D. The Class of (p,q)-spherical Distributions with an Extension of the Sector and Circle Number Functions. Risks 2017, 5, 40.

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