Wolf-Dieter Richter

Simulation of the p-generalized Gaussian distribution

In this paper, we introduce the p-generalized polar methods for the simulation of the p-generalized Gaussian distribution. On the basis of geometric measure representations, the well-known Box–Muller method and the Marsaglia–Bray rejecting polar method for the simulation of the Gaussian distribution are generalized to simulate the p-generalized Gaussian distribution, which fits much more flexibly to data than the Gaussian distribution and has already been applied in various fields of modern sciences. To prove the correctness of the p-generalized polar methods, we give stochastic representations, and to demonstrate their adequacy, we perform a comparison of six simulation techniques w.r.t. the goodness of fit and the complexity. The competing methods include adapted general methods and another special method. Furthermore, we prove stochastic representations for all the adapted methods.

Ellipses numbers and geometric measure representations

Abstract Ellipses will be considered as subsets of suitably defined Minkowski planes in such a way that, additionally to the well-known area content property A(r) = π (a,b) r 2, the number π (a,b) = abπ reflects a generalized circumference property U (a,b)(r) = 2π (a,b) r of the ellipses E (a,b)(r) with main axes of lengths 2ra and 2rb, respectively. In this sense, the number π (a,b) is an ellipse number w.r.t. the Minkowski functional r of the reference set E (a,b)(1). This approach is closely connected with a generalization of the method of indivisibles and avoids elliptical integrals. Further, several properties of both a generalized arc-length measure and the ellipses numbers will be discussed, e.g. disintegration of the Lebesgue measure and an elliptically contoured Gaussian measure indivisiblen representation, wherein the ellipses numbers occur in a natural way as norming constants.

Geometric and Stochastic Representations for Elliptically Contoured Distributions

A non Euclidean geometric measure representation for elliptically contoured distributions and a stochastic representation for corresponding random vectors are derived in a similar way as analogous representations were derived in Richter (2007)Richter (2009) for l n, p -symmetric distributions. The ball number function and the indivisiblen method of Cavalieri and Torricelli are extended to ellipsoids.

Probabilities and Large Quantiles of Noncentral Generalized Chi-Square Distributions

Exact values of probability integrals for noncentral generalized chi-square distributions are numerically evaluated based upon new geometric representation formulae for these distributions. Using iterative numerical methods exact quantiles can be calculated then. Explicit quantile approximation formulae are deduced from an asymptotic expansion for related probabilities of large deviations. Though this method is originally directed to the construction of starting values for determining exact large quantiles it is of benefit for simply approximating large quantiles and for obtaining quantiles from the central part of the distributions, too. The accuracy of the explicit asymptotic approximation method can be improved by combining it with the geometric measure representation formulae. Several numerical studies compare the present results with results of other authors available in the special case of the classical noncentral chi-square distribution. As an application, critical test points as well as power functions for expectation tests in elliptically contoured sample distributions are considered and certain problems of sensitivity and robustness type are discussed.

Convex and Radially Concave Contoured Distributions

Integral representations of the locally defined star-generalized surface content measures on star spheres are derived for boundary spheres of balls being convex or radially concave with respect to a fan in . As a result, the general geometric measure representation of star-shaped probability distributions and the general stochastic representation of the corresponding random vectors allow additional specific interpretations in the two mentioned cases. Applications to estimating and testing hypotheses on scaling parameters are presented, and two-dimensional sample clouds are simulated.

Exact distributions under non-standard model assumptions

A rather general method is presented of how to derive the exact distribution of a function of a random vector if the vector does not satisfy standard model assumptions.

Linear Combinations, Products and Ratios of Simplicial or Spherical Variates

The density level sets of the two types of measures under consideration are l 2, p -circles with p = 1 and p = 2, respectively. The intersection-percentage function (ipf) of such a measure reflects the percentages which the level set corresponding to the p-radius r shares for each r > 0 with a set to be measured. The geometric measure representation formulae in Richter (2009) is based upon these ipfs and will be used here for evaluating exact cdfs and pdfs for the linear combination, the product, and the ratio of the components of two-dimensional simplicial or spherically distributed random vectors.

Circle Numbers for Star Discs

The notion of a generalized circle number which has recently been discussed for 𝑙2,𝑝-circles and ellipses will be extended here for star bodies and a class of unbounded star discs.

A geometric approach to finite sample and large deviation properties in two-way ANOVA with spherically distributed error vectors

This article investigates a two-way ANOVA model with interactions assuming that the vector of error variables possesses a general spherically symmetric distribution instead of a multivariate normal one. Via a geometric approach we study a test for the usual hypothesis of non-interaction under this general assumption. Moreover, based on a certain geometric representation formula, we establish exponential large deviation rates of the least squares estimators in the above model for a specific class of spherical distributions.

Geometric Approach to Evaluating Probabilities of Correct Classification into two Gaussian or Spherical Categories

The problem of classification is considered within a linear model approach. Two competing hypotheses concerning one-dimensional expectations are reflected by so-called “hypothesis”, suitable subspaces of the model space. The distances between either the sample point or its projection onto the extended model space and the hypotheses spaces are equivalently used to decide between the competing hypotheses. The event of making a correct statistical decision is then described by suitable subsets of the sample space or of the extended model space, respectively. The probability of the sets is calculated by using geometric representation formulas for Gaussian or spherical distributions in the respective spaces. In this way we get the explicit numerical values for the probabilities of a correct classification.