Circle Numbers of Regular Convex Polygons

Richter, Wolf-Dieter; Schicker, Kay

doi:10.1007/s00025-016-0534-y

Cited by 7 publications

(9 citation statements)

References 18 publications

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“…For a geometric generalization of the multivariate exponential law, we refer to the class of regular simplicially contoured or l 1 -norm symmetric distributions studied in (Henschel and Richter 2002). Further results for p-spherical distributions with p from {1, 2, ∞} can be found in (Rachev and Rueschendorf 1991), (Kamiya et al 2008) and (Richter and Schicker 2016).…”

Section: Asymmetric (P Q)-spherical Generalization Of the Gauss-expomentioning

confidence: 99%

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

Richter

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.

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Section: Asymmetric (P Q)-spherical Generalization Of the Gauss-expomentioning

confidence: 99%

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

Richter

2017

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“…(b2) If P is additionally convex then For details on the application of the latter representation, we refer to [25]. This representation is used there and elsewhere to study ball numbers, circle numbers, and generalized uniform distributions on the boundaries of platonic bodies and regular convex polygons, respectively.…”

Section: Lemmamentioning

confidence: 99%

“…, − 1, are the same as those in the case of usual polar coordinates. For calculating the Jacobian, we refer to the proof of Theorem 3.1 in [25] that can be extended to the -dimensional case.…”

mentioning

confidence: 99%

Polyhedral Star-Shaped Distributions

Richter

Schicker

2017

Journal of Probability and Statistics

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A new method of probabilistic modelling of polyhedrally contoured sample clouds is presented and applied to statistical reasoning for a real dataset. Various representations of the new class of polyhedral star-shaped distributions are derived and basic properties of the moments as well as characteristic and moment generating functions of these distributions are studied. Along with location-scale transformations, estimating and hypothesis testing are dealt with.

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“…For a geometric generalization of the multivariate exponential law we refer to the class of regular simplicially contoured or l 1 -norm symmetric distributions studied in [4]. Further results for p-spherical distributions with p from {1, 2, ∞} can be found in [8], [6] and [18].…”

Section: Asymmetric (P Q)-spherical Generalization Of the Gauss-expomentioning

confidence: 99%

The Class of (p,q)-spherical Distributions

Richter¹

2017

Preprint

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For evaluating 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 upon the Gauss-Laplace law. The latter will be considered here as an element of the newly introduced family of (p, q)-spherical distributions. Based upon 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 like with elliptically contoured distributions and more general homogeneous star-shaped ones. This is demonstrated at hand of a generalization of the Box-Muller simulation method. En passant, we prove an extension of the sector and circle number functions.

show abstract

Circle Numbers of Regular Convex Polygons

Cited by 7 publications

References 18 publications

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

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

Polyhedral Star-Shaped Distributions

The Class of (p,q)-spherical Distributions

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