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

By Wolf-Dieter Richter

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

Topics: 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, Insurance, HG8011-9999
Publisher: MDPI AG
Year: 2017
DOI identifier: 10.3390/risks5030040
OAI identifier: oai:doaj.org/article:e9f7a2ddab414af8a016279fea68689f
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