Journal of Probability and Statistics

Volume 2017, Article ID 7176897, 23 pages

https://doi.org/10.1155/2017/7176897

## Polyhedral Star-Shaped Distributions

Institute of Mathematics, University of Rostock, Ulmenstr. 69, Haus 3, 18057 Rostock, Germany

Correspondence should be addressed to Wolf-Dieter Richter; ed.kcotsor-inu@rethcir.reteid-flow

Received 27 July 2016; Revised 2 November 2016; Accepted 3 November 2016; Published 14 February 2017

Academic Editor: Ramón M. Rodríguez-Dagnino

Copyright © 2017 Wolf-Dieter Richter and Kay Schicker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

One of nowadays challenges of statistical modelling is the construction of flexible multivariate probability distributions given a dataset. In [1], disparities in premature mortality between high- and low-income US counties are dealt with on the basis of descriptive statistics. For a subsequent step of statistical reasoning, a method of constructing a suitable probabilistic model is needed. Analyzing the correlation between mortality and income cannot be done in a common linear regression model because of absence of homoscedasticity. The present paper provides therefore a new method of constructing flexible multivariate distribution laws being well adapted to practical problems characterized by polyhedral contours of their sample clouds. Moreover, we will particularly suggest a specific model and further statistical reasoning for the premature mortality-income data mentioned above.

Several basic methods and results from the research area of constructing multivariate distributions are surveyed, for example, in [2] and the numerous papers mentioned there. Multivariate densities with given contours were already introduced in [3] and further studied in [4, 5]. Constructing star-shaped distributions on using Minkowski functionals and generalized uniformly distributed random vectors on generalized spheres, proving geometric measure representations of such distributions and stochastic representations of correspondingly distributed random vectors are to be found in [6]. Constructions and representations in the special cases of norm and antinorm contoured distributions are recently dealt with in [7]. Numerous applications of geometric measure representations are surveyed in the last mentioned two papers and in [8]. In [9] more recent applications to extreme values are presented. More general order statistics are dealt with elsewhere.

The aim of the present paper is to deal with another important special case of star-shaped distributions where the contours are the topological boundaries of polyhedra.

Polyhedral star-shaped distributions can be considered being a subclass of the class of star-shaped distributions. Before studying specific properties of polyhedral star-shaped distributions, we start therefore with a short introduction to the general theory of star-shaped distributions. To this end, we follow [6].

Let denote a star body, that is, a nonempty and compact star-shaped set being equal to the closure of its interior and having the origin in its interior. The functional defined byis known as the Minkowski functional of . Here, we assume that is positive homogeneous; that is, , , and consider and its boundary being the star ball and star sphere of Minkowski radius , respectively. Since unambiguously defines the considered star ball, it is possible to study subclasses of star bodies by specifying . One can choose, for example, to be a norm or an antinorm. For the latter notion we refer to [10]. Here, specific representations of are considered later in this paper, if denotes a star-shaped polyhedron. For simplicity, consideration will be restricted throughout this paper to star bodies having the following property.

A countable collection of pairwise disjoint cones with vertex being the origin and is called a fan. Let , , where denotes the Borel--field in , and . In what follows, the star body and a set are chosen such that for every the set is well defined and for every there is uniquely determined satisfying .

A function satisfying the assumptions where is called a density generating function (dgf),a star-shaped density and its contour defining star body. Such densities are studied in [3, 6, 11]. A probability measure having the density will be denoted by . Note that the normalizing constant allows the representationwhere denotes the Lebesgue measure in . For examples of density generating functions, see [12] or Table 1. Moreover, the definition of specific star-shaped densities appears already in earlier work where is prespecified. If is the Euclidean-norm, one considers the class of spherically symmetric distributions; see [12–14] and many other contributions. The class of convex contoured -symmetric distributions is considered if is the -norm, . This class of distributions was introduced in [15] and studied, for example, in [16–19]. In [20] stochastic and geometric representations are derived also for the case which can be called according to [10] the antinorm contoured case. General norm contoured distributions in are studied in [8] by choosing in terms of a norm in . Minkowski functionals of ellipsoids and even -generalized ellipsoids are used in the (re-)construction of common and -generalized elliptically contoured distributions in [21] and in [6], respectively.