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Journal of Probability and Statistics
Volume 2017 (2017), Article ID 7176897, 23 pages
https://doi.org/10.1155/2017/7176897
Research Article

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

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 [1214] 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 [1619]. 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.

Table 1: Density generating functions, normalizing constants, and moments of .

The introduction of as generalized radius functional is closely connected with the notion of generalized surface content measure which turns out to be a suitably chosen non-Euclidean surface content measure on a star sphere. These notions and their properties are very useful in the consideration of nonspherical distributions; see [6, 7, 20, 21]. Here, we recall the local definition of a generalized surface content measure on star spheres from [22], noting that there exists an equivalent integral approach introduced for various special classes of star bodies in [6, 7, 20, 21]. For , we introduce the central projection cone and the star sector of star radius , . The star-generalized surface measure is then defined byThe star-generalized uniform probability distribution on the Borel--field is defined as . With these notations, we can recall the geometric measure representation formula of for every ,where denotes the star intersection-proportion function (ipf) of the set . Furthermore, a star-shaped distributed random vector allows the stochastic representationwhere and are stochastically independent, , and follows the density , . Furthermore, we recall from [7] integral representations of , , where denotes the Borel--field on the upper half-sphere of , if is a norm and an antinorm, respectively. It holdswhere with , denotes the dual norm of , if is a norm, and denotes the Minkowski functional of the antipolar set of if is an antinorm with being an element of the class of antinorm balls AN1 defined in [7]. Here, is the outer normal vector to at and is the inner normal vector to at , respectively. For particular representations of and if is a norm or antinorm, respectively, generated by a star-shaped polyhedron, we refer already here to Theorem 6 (d) and (e) below. In this context, we will say that belongs to the class of antinorm balls AN2 if is an antinorm and is a star-shaped polyhedron. We notice that convex geometry and certain types of polytopes play also a basic role in [23] where max-stable distributions and all norms that give rise to strictly max-stable distributions are studied.

The paper is now organized as follows. We present the Minkowski functionals of star-shaped polyhedra in various kinds in Section 2 and derive corresponding representations of the polyhedral star-shaped surface and uniform measures in Section 3. This allows an extension of the ball number function in Section 3.2. The new classes of general polyhedral star-shaped distributions and specific -star contoured distributions are considered then in Sections 4.1 and 4.2, respectively. Moments, characteristic, and moment generating functions are studied in Section 5, and simulation with a rejection method is discussed in Section 6. Location-scale transformations considered in Section 7 give rise to considering estimating of parameters and testing of hypothesis in Section 8. In Section 8, moreover, we present the probabilistic modelling and further statistical reasoning for the premature mortality-income data mentioned at the beginning of this introduction. We discuss some open problems in Section 9. Appendix mainly shows how various theoretical results apply in one and the same situation.

2. Minkowski Functionals of Star-Shaped Polyhedra

In this section, we specify the representation of if the star ball is a star-shaped polyhedron. Let be a star-shaped polyhedron. We call a convex polyhedral fan for if is for every , a convex polyhedron. Due to the considerations of convex polyhedra in the broad literature of convex geometry, it is possible to represent a convex polyhedron in two different but equivalent ways; see [24]. A convex polyhedron can be given by the set of its vertices , where denotes the convex hull of the point set . Alternatively, it is possible to consider a convex polyhedron as the intersection of suitably chosen closed half-spaces. In this case, there exist a matrix and a vector such that where “” is declared componentwise. The following proposition concludes that star-shaped polyhedra allow analogous representations.

Proposition 1. If is a star-shaped polyhedron and a convex polyhedral fan for then always allows each of the following two equivalent representations.(a)There exist integer numbers and points   from , such that and (b)There exist matrices and vectors with positive real components such that

The following lemma is basic for proving the main results of this paper. It makes use of both representations in Proposition 1. For any and , let be the indicator function defined by and .

Lemma 2. Let and be as in Proposition 1. (a)With notation as in , can be equivalently represented as follows:(a1).(a2).(a3), where denotes the solution of the following linear optimization problem (LOP): minimize subject to the conditions and , , where , .(a4)If additionally for every , , is an -dimensional simplex then where denotes the -dimensional vector of ones, , and reads componentwise.(b1)With notation as in , where is assumed to have facets and for every , , there exist reals , , such that allows the representation (b2)If   is additionally convex then with , where the quantities and are equal to those in and , is declared componentwise.

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.

Proof. Representation (a1) follows immediately by applying the definition of in (1) and . Representation (a2) follows then by applying (a1) and the well-known representation for a point set in . From (a2), it follows that has to be minimal where and has to be represented by the linear combination . From this, (a3) follows. We derive now representation (a4) from (a3). If all are -dimensional simplizes then and is uniquely determined since is built up by affinely independent points thus satisfying . In turn, the minimum of can be calculated directly by . Furthermore, if a given vector is an element of the simplex , all have to be nonnegative in this case. Thus the vector satisfies iff all components of are nonnegative, yielding the definition of .
Representation (b1) can be proved as follows: if , are the facets of , each is a subset of the boundary of an -dimensional, closed half-space. Thus, there exist reals , , such that From this, it follows Representation (b2) follows by applying and using that

3. Polyhedral Generalized Surface Content Measure

3.1. Specific Representations

In this section, specific representations of the star-generalized surface content measure are considered in case that is a star-shaped polyhedron and its boundary. Prior to this, we consider a generalized polar coordinate transformation allowing one of the specific representations derived later on. For another powerful application of these generalized polar coordinates, we refer already here to Section 5.2.

Definition 3. Let be a star-shaped polyhedron. The polyhedral star-shaped polar coordinate transformation , , is defined by where .

We will later also use the notation if .

Lemma 4. The map is almost one-to-one, for , its inverse is given by and its Jacobian is

Proof. It holdsand the relations of ,  , are the same as those in the case of usual polar coordinates. For calculating the Jacobian, we refer to the proof of Theorem in [25] that can be extended to the -dimensional case.

Let us denote the cdf of a random vector by .

Theorem 5. Let be the random polyhedral star-shaped coordinates of , where ; then and are stochastically independent, , and with , and

Proof. Applying the described method in Remark in [6] for elliptically contoured distributions analogously here yields the representation of . Furthermore, for every , where . Thus with

Let us denote by the restriction of to the case and, analogously, by the inverse of the map . Now, the following representations of can be proved.

Theorem 6. Let be a star-shaped polyhedron and . (a) satisfies the polyhedral star-shaped polar coordinate representation (b1)With notation as in and Lemma 2 (b1), satisfies the star-spherical coordinate representation (b2)Let additionally denote the volume in , where , if , ; then satisfies the facet-content representation where . Note that is proportional to the Lebesgue measure in in every sector , .(c)Let additionally , be -dimensional simplizes and and , represented according to . Then there exist integers such that and , , where are -dimensional simplizes. Then satisfies the simplicial representation where(d)If additionally denotes a norm and is the outer normal vector to the norm sphere at the point , then, with notations as in , satisfies the dual norm representation(e)If additionally denotes an antinorm and and is the inner normal vector to the antinorm sphere at the point , then with notations as in the Minkowski functional of the antipolar set of defined by , and satisfies the (antipolar set) representation

Proof. Since (5) yields (a). Analogously, where, for every , , is the value of the Jacobian of the star-spherical coordinate transformation; see [6]. Since , can be represented by star-spherical coordinates as it follows by applying Lemma in [6] that . Thus, (5) yields (b1). Considering part (b2), it holds , where every , is a convex cone with base and cusp in the origin. Thus , where denotes the Euclidean length of the height of . According to Remark in [6], it holds . Thusand (b2) is proved. Considering part (c), it holds and since it follows Considering part (d) and applying (8), it remains to prove that . According to [26] it is well known that equals the support function of ; thus , , and it follows .
Considering part (e) we can use the results from Lemma in [6], according to which it holds With notations as in , it follows for every that and . Thus, for every , Since the antisupport function of with respect to is defined by
and it is shown in Lemma in [7] that , part (e) is proved.

According to [7] it is possible to represent the antipolar set of a by where denotes the Euclidean unit sphere in . For an illustration, we refer to Section 4.2, where antipolar sets of -stars are considered. For an application of certain representations of Theorem 6, we refer to Example A.1 in the Appendix.

3.2. Extension of the Ball Number Function

The circumference and area content properties of Euclidean circles which motivate the generalization of the circle number have been discussed to a certain extent first in [27] for -circles. These considerations were later extended to ellipses and general star-circles. Related multivariate studies started by introducing the generalized surface and volume properties of -balls and were followed up in [6, 21]. General ball numbers being values attained by the ball number function are defined in the literature, and it is also stated there as a challenging problem to extend the ball number function to as many as possible further classes of generalized balls. In this respect, an extension of the ball number function to the class of regular convex polygons can be found in [25], and an extension to platonic bodies is considered elsewhere by the authors. Furthermore, in Section 7 of [25], several other possible extensions of the circle number function one could think about are discussed.

Note that the equation reflects a certain generalization of the method of indivisibles of Cavalieri and Torricelli and that the ratios not only do coincide but are even independent of . This motivates the following definition.

Definition 7. The polyhedral star ball number is defined by .
Note that . For concrete values of in the special case of regular convex polygons, see [25]. Ball numbers of platonic bodies are dealt with elsewhere. Apart from this extension of the domain of the ball number function, will be used in this paper as kind of normalizing constant for polyhedral star-shaped contoured distributions.

3.3. Polyhedral Star-Shaped Generalized Uniform Distributions

The notion of a polyhedral generalized surface content measure makes it possible to define the polyhedral star-shaped generalized uniform distribution on asNote that plugging in any of the representations of Theorem 6 into (42) yields the polyhedral star-shaped polar coordinate, star-spherical coordinate, and facet-content, simplicial and dual norm representations of the generalized uniform distribution, respectively. A numerical example of the application of can be found in Example A.1 in the Appendix.

Let be a probability space and a random vector being defined on and taking values in . Assume further that follows the uniform probability distribution on ; that is, , for , and put , where division is defined componentwise. The proof of the following result can be done analogously to that in case of platonically generalized uniform distributions on platonic spheres.

Theorem 8. The random vector follows the polyhedral star-shaped generalized uniform distribution on .

4. Polyhedral Star-Shaped Distributions

4.1. Representations

The results in the previous section can be used for representing general polyhedral star-shaped distributions. Doing this, we follow the considerations in [68, 12, 20, 21, 28], where stochastic representations like (7) were repeatedly exploited.

A random vector is said to follow a polyhedral star-shaped distribution if there exists a random variable such that satisfies the stochastic representationwhere is polyhedral star-shaped generalized uniformly distributed on and and are stochastically independent. The set of all polyhedral star-shaped distributions on will be denoted by and its subset of continuous distributions by Applying (6) and the representations of Theorem 6, we can state the following specific geometric measure representation formulae of continuous polyhedral star-shaped distributions. For the application of geometric measure representation formulae of other classes of distributions, we refer to [2830].

Proposition 9. Let the assumptions from Theorem 6 be fulfilled and . Then allows correspondingly (a)polyhedral star-shaped polar coordinate representation (b1)star-spherical coordinate representation (b2)facet-content representation if additionally where and , if ,(c)simplicial representation (d)dual norm representation (e)(antipolar set) representation (f)sector measure representation Note that for the sector measure also the notion of cone measure is used; see [17, 31]

Finally note that if then we write (43) aswhere the nonnegative random variable is independent of .

4.2. The Class of -Star Contoured Distributions in

The polyhedral star-shaped distributions considered in this section may be alternatively norm or antinorm contoured. For simplicity, consideration is restricted here to the two-dimensional case. Let be a (vertices) driven star-shaped polygon defined for arbitrary by the vertices , , , , , , , and . We call a -star. For an illustration of different -stars, see Figure 1. Note that a -star appears to be convex if and radially concave if . For representing the Minkowski functional of a -star, it is convenient to make use of the fans and if or , respectively; see again Figure 1. If thenand if then Surprisingly enough, it follows that for every . This result enables us to introduce the -star contoured densities Note that since the normalizing constant applies by (4). If then denotes a norm. Thus, applying Theorem 6 (d), we can represent of a set by the dual norm representation as where . Note that further representations of the star-generalized surface content measure can be found in [7] for a special class of radially concave star bodies and in [25] for regular polygons, especially in the case that these polygons have an odd number of vertices and thus are not symmetric.

Figure 1: -stars in case (a) and in case (b).

If then denotes an antinorm. Thus, applying Theorem 6 (e), we can represent of a set by the (antipolar set) representation where the sectors , , are equivalent to those in Example A.2, and , , are equivalent to the entries of the matrix in Example A.2, , and the antipolar set of is the -ball of Minkowski radius .

5. Expectations

5.1. Moments of Polyhedral Star-Shaped Distributions

Let be uniformly distributed on a star-shaped polyhedron . Such vector has dgf . Since the density of isWith , , the vector of moments of order is defined as . Mixed moments of order are The numerical computation of these moments of requires the integration of polynomials over star-shaped polyhedra. To solve this computational problem, especially in higher dimensions, one can use the software package “LattE Integrale” which computes integrals of polynomials over convex polyhedra. The polyhedra can be given by or , and the results are exact if the entries (vertices in case of or and in case of ) are rational numbers; see [3234]. Note that star-shaped polyhedra can always be divided into disjoint convex polyhedra , say. Moreover, can also be computed by integrating the function over .

Theorem 10. Let be a star-shaped polyhedron, uniformly distributed on and . (a)If the moment exists then .(b)If exists then, for all , with , (c)In case of existence, the covariance matrix of allows the representation

Proof. By the stochastic representation in (52), we have . According to Theorem 8, where and are stochastically independent, follows the density function , andSince it follows (a). Parts (b) and (c) can be proved analogously.

For basic types of dgf and representations of the moments of , we refer to [12] and to Table 1.

5.2. Characteristic Functions

Integral representations of characteristic functions of arbitrary star-shaped distributed random vectors are derived in [6]. Specific representations are proved for spherically distributed random vectors in [35], for -symmetrically distributed random vectors in [36] and for -symmetrically distributed random vectors in [37]. We present a specific integral representation for continuous polyhedral star-shaped distributed random vectors by applying the polyhedral star-shaped polar coordinates to the arbitrary representations from [6]. This change of the coordinate system is very fruitful for the visualization of characteristic functions of continuous polyhedral star-shaped distributions by numerical methods; see Figures 2 and 3.