Geometric and Stochastic Representations for Elliptically Contoured Distributions

Richter, Wolf-Dieter

doi:10.1080/03610926.2011.611320

Cited by 16 publications

(44 citation statements)

References 7 publications

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“…Moreover, we emphasize the remarkable differences between convex and radially concave cases. In the sequel, in a certain analogy to what was done in [9][10][11][12] for various situations, we define the (p, q)-generalized arc length measure AL p,q on the Borel σ-field B(S p,q ) of S p,q . To this end, for an arbitrary set A ∈ B(S p,q ), let us call…”

Section: The (P Q)-arc Length Measure and Dynamic Geometric Disintegmentioning

confidence: 99%

“…If Y is a uniformly on (0, 1)-distributed random variable then Y pq/(p+q) follows the distribution having the density in (12). On the one hand, the (p, q)-spherical uniform probability distribution is singular with respect to the Lebesgue measure in the two-dimensional Euclidean space R 2 , but, on the other hand, it is absolutely continuous with respect to the Lebesgue measure on the real line.…”

Section: The (P Q)-spherical Uniform Distributionmentioning

confidence: 99%

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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.

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Section: The (P Q)-arc Length Measure and Dynamic Geometric Disintegmentioning

confidence: 99%

Section: The (P Q)-spherical Uniform Distributionmentioning

confidence: 99%

The Class of (p,q)-spherical Distributions

Richter¹

2017

Preprint

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“…When modeling Lymphoma data, [35] analyze sample clouds of points, see Figures 2 and 3, which might be interpreted as mixtures of densities having contours in part looking similar like that in [36] where flow cytometric data, Australian Institute of Sport data and Iris data are analyzed, or like that of certain skewed densities as they were (analytically derived and) drawn in [37]. In a similar manner, Figures 2 and 5 in [20] indicate that mixtures of different types of star-shaped distributions might be suitable for modeling residuals of certain stock exchange indices.…”

Section: Applicationsmentioning

confidence: 99%

Estimation of Star-Shaped Distributions

Liebscher

Richter

2016

Risks

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Scatter plots of multivariate data sets motivate modeling of star-shaped distributions beyond elliptically contoured ones. We study properties of estimators for the density generator function, the star-generalized radius distribution and the density in a star-shaped distribution model. For the generator function and the star-generalized radius density, we consider a non-parametric kernel-type estimator. This estimator is combined with a parametric estimator for the contours which are assumed to follow a parametric model. Therefore, the semiparametric procedure features the flexibility of nonparametric estimators and the simple estimation and interpretation of parametric estimators. Alternatively, we consider pure parametric estimators for the density. For the semiparametric density estimator, we prove rates of uniform, almost sure convergence which coincide with the corresponding rates of one-dimensional kernel density estimators when excluding the center of the distribution. We show that the standardized density estimator is asymptotically normally distributed. Moreover, the almost sure convergence rate of the estimated distribution function of the star-generalized radius is derived. A particular new two-dimensional distribution class is adapted here to agricultural and financial data sets.

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“…, ) T ∈ R ν and whose remaining columns are equal the zero vector. Each such representation can be interpreted as a measure-of-cone representation of a certain skewed ln−ν,p-symmetric distribution generalizing the results for two-and multivariate skewed elliptically contoured distributions in [16] and [34], respectively. Let us emphasize at this point that, on the one hand, the sets (x , .…”

Section: Introductionmentioning

confidence: 99%

“…We remark that extensions of this basic notion to ellipsoids, norm and antinorm spheres as well as to more general star spheres are given in [31][32][33] An n-dimensional random vector X : Ω → R n de ned on a probability space (Ω, A, P) and having the pdf…”

Section: Introductionmentioning

confidence: 99%

Exact distributions of order statistics from l_n,p-symmetric sample distributions

Müller

Richter

2017

Dependence Modeling

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Abstract:We derive the exact distributions of order statistics from a nite number of, in general, dependent random variables following a joint ln,p-symmetric distribution. To this end, we rst review the special cases of order statistics from spherical as well as from p-generalized Gaussian sample distributions from the literature. To study the case of general ln,p-dependence, we use both single-out and cone decompositions of the events in the sample space that correspond to the cumulative distribution function of the kth order statistic if they are measured by the ln,p-symmetric probability measure. We show that in each case distributions of the order statistics from ln,p-symmetric sample distribution can be represented as mixtures of skewed ln−ν,p-symmetric distributions, ν ∈ { , . . . , n − }.

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Geometric and Stochastic Representations for Elliptically Contoured Distributions

Cited by 16 publications

References 7 publications

The Class of (p,q)-spherical Distributions

The Class of (p,q)-spherical Distributions

Estimation of Star-Shaped Distributions

Exact distributions of order statistics from l_n,p-symmetric sample distributions

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Geometric and Stochastic Representations for Elliptically Contoured Distributions

Cited by 16 publications

References 7 publications

The Class of (p,q)-spherical Distributions

The Class of (p,q)-spherical Distributions

Estimation of Star-Shaped Distributions

Exact distributions of order statistics from ln,p-symmetric sample distributions

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Product

Resources

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Exact distributions of order statistics from l_n,p-symmetric sample distributions