Prof. Wolf-Dieter Richter
University of Rostock, Germany
Title: Distributions with different margins
Abstract:
Since a long time, the big class of spherical distribution laws plays an important role
in applied mathematics, in particular in statistics and multivariate distribution theory,
see e.g. Fang, Kotz and Ng (1990). Due to this theory, stochastic representations of
correspondingly distributed random vectors are an important analytical tool for mathematically analyzing many statistics and their distributions. The central role is played
in spherical distribution theory by the uniform probability law on the Euclidean unit
sphere.
All one-dimensional marginal distributions of a multivariate spherical distribution are
identical. There is, however, a need for considering multivariate distribution classes
allowing for di fferent marginal distributions. An example of such type appears as a
high risk limit law in Balkema and Embrechts (2007) and is discussed for dimension
two as a particular case of the class of (p; q)-spherical distributions in Richter (2017). A
specifi c notion of a non-Euclidean arc-length measure serves as the basis for constructing
a generalized uniform distribution on (p; q)-circles.
Here, we go a next step and extent the latter distribution class to dimension three.
From the technical point of view, the key point for introducing the new stochastic representation is the suitable adoption of the notion of surface content for defining a generalized uniform distribution on a generalized sphere. This is done by extending the
non-Euclidean arc-length measure on (p; q)-circles to a generalized surface content measure in the present three-dimensional situation.
Biography:
Wolf-Dieter Richter is an Emeritus Professor at the Institute of Mathematics at the
University of Rostock. His main areas of research are Mathematical Statistics and Exact
and Asymptotic Distribution Theory, namely generalized central and non-central
Chisquare, Student- and Fisher- distributions, non-linear regression, exact and asymptotic
distributions of likelihood-ratio classi cation rules, quantile approximation, simulation,
skewness-kurtosis adjusted decisions, Kullback-Leibler life time testing, geometric
and stochastic representations of star-shaped (especially convex and radially concave
as well as polyhedral contoured), skewed and directional distributions, exact statistical
distributions under non-standard assumptions, statistics in p-generalized elliptically contoured
sample distributions, limit theorems in multivariate moderate and large deviation
theory as well as scale mixtures and generalized stochastic processes. He has a Ph.D.
and a Habilitation in Mathematics at Technical University of Dresden, is a Fellow of
the Alexander von Humboldt-Stiftung, is the supervisor of ten Ph.D. students, is an ISI
Elected Member and an Associate Editor of the Journal of Statistical Distributions and
Applications. He is a reviewer for several mathematical journals, had research stays in
Vilnius, St.Petersburg, Moscow, Novosibirsk, Aarhus, Providence, Hong Kong, Warsaw,
Sao Paulo, Santiago de Chile, Vasa, Merida, Guanajuato, Valparaiso and Nottingham
and gave talks at many universities and conferences. In his work, he and his numerous
co-workers developed and combined analytical and geometric methods for analyzing
distributions, introduced antinorms and semi-antinorms, introduced the anti-support
function, introduced new trigonometric functions and coordinates for making his new
geometric disintegration method an efficient tool of mathematical work, established non-
Euclidean uniform distributions on generalized spheres, and the ball number function
whose values are normalizing constants of generating functions of multivariate densities.
Wolf-Dieter Richter's Erdoes number is three and eight steps back in his path of
dissertation advisors there is Pafnuty Lvovich Chebyshev.