# Polyhedral Star-Shaped Distributions

Authors: Wolf-Dieter Richter; Kay Schicker
Journal: Journal of Probability and Statistics
 (2017)

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## 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.


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## Body

## 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].LetK⊂Rn 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 0n in its interior. The functional hK:Rn→[0,∞) defined by(1)hKx=inf⁡r>0:x∈rK,x∈Rnis known as the Minkowski functional of K. Here, we assume that hK is positive homogeneous; that is, hK(λx)=λhK(x),  λ>0, and consider K(r)=rK={x∈Rn:hK(x)≤r} and its boundary S(r)=rS={x∈Rn:hK(x)=r} being the star ball and star sphere of Minkowski radius r>0, respectively. Since hK unambiguously defines the considered star ball, it is possible to study subclasses of star bodies by specifying hK. One can choose, for example, hK to be a norm or an antinorm. For the latter notion we refer to [10]. Here, specific representations of hK are considered later in this paper, if hK denotes a star-shaped polyhedron. For simplicity, consideration will be restricted throughout this paper to star bodies having the following property.A countable collectionF={C1,C2,…} of pairwise disjoint cones Cj with vertex being the origin 0n and Rn=⋃jCj is called a fan. Let Sj=S∩Cj,  Sj∩Bn=BS,j, where Bn denotes the Borel-σ-field in Rn, and BS=σ{BS,1,BS,2,…}. In what follows, the star body K and a set A∈BS are chosen such that for every j the set (2)GA∩Sj=ϑ∈Rn-1:∃ηwithθ=ϑT,ηT∈A∩Sjis well defined and for every ϑ=(ϑ1,…,ϑn-1)T∈G(A∩Sj) there is uniquely determined η>0 satisfying hK((ϑ1,…,ϑn-1,η)T)=1.A functiong:R+→R+ satisfying the assumptions 0<I(g)<∞ where I(g)=∫0∞rn-1g(r)dr is called a density generating function (dgf),(3)φg,Kx=Cg,KghKx,x∈Rna star-shaped density and K its contour defining star body. Such densities are studied in [3, 6, 11]. A probability measure having the density φg,K will be denoted by Φg,K. Note that the normalizing constant C(g,K) allows the representation(4)Cg,K=1nμKIg,where μ denotes the Lebesgue measure in Rn. 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 hK is prespecified. If hK is the Euclidean-norm, one considers the class of spherically symmetric distributions; see [12–14] and many other contributions. The class of convex contoured ln,p-symmetric distributions is considered if hK is the ln,p-norm, p≥1. 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 R2 are studied in [8] by choosing hK in terms of a norm in R2. Minkowski functionals of ellipsoids and even p-generalized ellipsoids are used in the (re-)construction of common and p-generalized elliptically contoured distributions in [21] and in [6], respectively.Table 1
Density generating functions, normalizing constants, and moments ofRg.
Type g ( r ) I ( g ) E ( R g ) E ( R g 2 ) E(Rgk) Kotz r 2 M - 2 e - β r 2 γ, M,β,γ∈(0,∞) Γ 2 M + n - 2 / 2 γ 2 γ β 2 M + n - 2 / 2 γ Γ 2 M + n - 1 / 2 γ β 1 / 2 γ Γ 2 M + n - 2 / 2 γ Γ 2 M + n / 2 γ β 1 / γ Γ 2 M + n - 2 / 2 γ Γ 2 M + n + k - 2 / 2 γ β k / 2 γ Γ 2 M + n - 2 / 2 γ  Multinormal e - r 2 / 2 Γ ( n / 2 ) 2 1 - n / 2 Γ n + 1 / 2 2 Γ n / 2 n Γ n + k / 2 2 k / 2 Γ n / 2  Pearson VII 1 + r 2 m - M, M>n2,  m>0 Γ ( n / 2 ) Γ ( M - n / 2 ) 2 Γ ( M ) m - n / 2 Γ n + 1 / 2 Γ M - n + 1 / 2 m Γ n / 2 Γ M - n / 2 m n 2 M - n - 2 Γ n + k / 2 Γ M - n + k / 2 m - k / 2 Γ n / 2 Γ M - n / 2  Multivariatet 1 + r 2 m - m / 2 - n / 2, m∈N>0 Γ ( n / 2 ) Γ ( m / 2 ) 2 Γ ( m + n / 2 ) m - n / 2 Γ n + 1 / 2 Γ m - 1 / 2 m Γ n / 2 Γ m / 2 m n m - 2 Γ n + k / 2 Γ m - k / 2 m - k / 2 Γ n / 2 Γ m / 2  Multivariate Cauchy 1 + r 2 - n + 1 / 2 π Γ ( n / 2 ) 2 Γ ( n + 1 / 2 ) — — —  Pearson II I [ 0,1 ] ( r ) 1 - r 2 m, m>-1 Γ ( n / 2 ) Γ ( m + 1 ) 2 Γ ( n / 2 + m + 1 ) Γ n + 1 / 2 Γ n / 2 + m + 1 Γ n / 2 + m + 3 / 2 Γ n / 2 n 2 m + n + 2 Γ n + k / 2 Γ n / 2 + m + 1 Γ n + k / 2 + m + 1 Γ n / 2  Uniform I 0,1 r 1 n n n + 1 n n + 2 n n + k n n + k X XThe introduction ofhK 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 A∈BS, we introduce the central projection cone CPC(A)={x∈Rn:x/hK(x)∈A} and the star sector of star radius ρ>0, sector(A,ρ)=CPC(A)∩K(ρ). The star-generalized surface measure is then defined by(5)OSA=f′ρ,wherefρ=μsectorA,ρ,A∈BS.The star-generalized uniform probability distribution on the Borel-σ-field BS is defined as ωS(A)=OS(A)/OS(S). With these notations, we can recall the geometric measure representation formula of Φg,K(B) for every B∈Bn,(6)Φg,KB=Cg,K∫0∞rn-1grOS1rB∩Sdr=Cg,KOSS∫0∞rn-1grFSB,rdr,where r→FS(B,r)=ωS1/rB∩S denotes the star intersection-proportion function (ipf) of the set B. Furthermore, a star-shaped distributed random vector Y~Φg,K allows the stochastic representation(7)Y=dRgUS,where Rg and US are stochastically independent, US~ωS, and Rg follows the density f(r)=rn-1g(r)/I(g), r>0. Furthermore, we recall from [7] integral representations of OS(A), A∈BS+, where BS+ denotes the Borel-σ-field on the upper half-sphere of S, if hk is a norm and an antinorm, respectively. It holds(8)OSA=∫GAhK∗Nϑdϑ,OSA=∫GAhK∘N∘ϑdϑ,where G(A)={ϑ∈Rn-1:∃η=η(ϑ) with (ϑT,η)T∈A}, hK∗ denotes the dual norm of hK, if hK is a norm, and hK∘ denotes the Minkowski functional of the antipolar set K∘ of K if hk is an antinorm with K being an element of the class of antinorm balls AN1 defined in [7]. Here, N(ϑ) is the outer normal vector to S at (ϑT,η)T and N∘(ϑ) is the inner normal vector to S at (ϑT,η)T, respectively. For particular representations of hK∗ and hK∘ if K 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 K belongs to the class of antinorm balls AN2 if hK is an antinorm and K=P 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 Section2 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 w-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 ofhK if the star ball K is a star-shaped polyhedron. Let P⊂Rn be a star-shaped polyhedron. We call F={C1,…,Cm} a convex polyhedral fan for P if P∩Cj is for every j, j∈{1,…,m} 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 P⊂Rn can be given by the set of its vertices {p1,…,pl}, VPP=convp1,…,pl,where convM denotes the convex hull of the point set M. 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 A∈Rk×n and a vector b∈Rk such that HPP=x∈Rn:Ax≤b,where “≤” is declared componentwise. The following proposition concludes that star-shaped polyhedra allow analogous representations.Proposition 1.
IfP⊂Rn is a star-shaped polyhedron and F={C1,…,Cm} a convex polyhedral fan for P then P always allows each of the following two equivalent representations.(a)
There exist integer numbersl1,…,lm and points pj,1,…,pj,lj  from Rn, j∈{1,…,m} such that P∩Cj=conv{0n,pj,1,…,pj,lj} and SVPP=⋃j=1mconv0n,pj,1,…,pj,lj.(b)
There exist matricesAj∈Rkj×n and vectors bj∈R+kj with positive real components such thatSHPP=⋃j=1mx∈Rn:Ajx≤bj.The following lemma is basic for proving the main results of this paper. It makes use of both representations in Proposition1. For any C⊂Rn and x∈Rn, let IC:Rn→{0,1} be the indicator function defined by (9)ICx=1,ifx∈C,0,otherwise,posx={αx∣α≥0} and posC=conv⋃x∈Cposx=conv⋃α≥0αC.Lemma 2.
LetP and F be as in Proposition 1. (a)
With notation as inSVP, hP can be equivalently represented as follows:
(a1)
hP(x)=∑j=1mICjxinf⁡r>0:x∈conv{0n,rpj,1,…,rpj,lj}.(a2)
hP(x)=∑j=1mICjxinf⁡r>0:x∈∑i=1ljλipj,i,λi≥0,∑i=1ljλi=r.(a3)
hP(x)=∑j=1mICj(x)hP(j)(x), where hP(j)(x) denotes the solution of the following linear optimization problem (LOP): minimize λ1+⋯+λlj subject to the conditions A(j)(λ1,…,λlj)T=x and λi≥0, i∈{1,…,lj}, where A(j)=(pj,1,…,pj,lj)∈Rn×lj, j∈{1,…,m}.(a4)
If additionally for everyj, j∈{1,…,m}, P∩Cj is an n-dimensional simplex then (10)hPx=∑j=1mICjx1nTAj-1x,where1n=(1,…,1)T denotes the n-dimensional vector of ones, Cj={x∈Rn:(A(j))-1x≥0}, and ≥ reads componentwise.(b1)
With notation as inSHP, where P is assumed to have m facets Hj and Cj=posHj for every j, j∈{1,…,m}, there exist reals aj1,…,ajn, j=1,…,m, such that hP allows the representation (11)hPx=∑j=1mICjxaj1x1+aj2x2+⋯+ajnxn,x=x1,…,xnT∈Rn.(b2)
IfP is additionally convex then (12)hPx=max⁡A1′x,…,Am′x,withA′=(A1′,…,Am′)T=(A1/b1,…,Am/bm)T, where the quantities A=(A1,…,Am)T and b=(b1,…,bm)T are equal to those in HP and Aj/bj, j∈{1,…,m} 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 ofhP in (1) and SVP. Representation (a2) follows then by applying (a1) and the well-known representation (13)convx1,…,xl=∑i=1lλixi:λi≥,∑i=1lλi=1for a point set {x1,…,xl} in Rn. From (a2), it follows that ∑i=1ljλi has to be minimal where λi≥0 and x∈Rn has to be represented by the linear combination ∑i=1ljλipj,i. From this, (a3) follows. We derive now representation (a4) from (a3). If all P∩Cj are n-dimensional simplizes then A(j)∈Rn×n and A(j)(λ1,…,λn)T=x is uniquely determined since A(j) is built up by affinely independent points thus satisfying det(A(j))≠0. In turn, the minimum of ∑i=1nλi can be calculated directly by 1nT((A(j))-1x). Furthermore, if a given vector x is an element of the simplex P∩Cj, all λ1,…,λn have to be nonnegative in this case. Thus the vector λ=(A(j))-1x satisfies x∈(P∩Cj) iff all components of λ are nonnegative, yielding the definition of ICj.
Representation (b1) can be proved as follows: ifHj, j=1,…,m are the facets of P, each Hj is a subset of the boundary of an n-dimensional, closed half-space. Thus, there exist reals aj1,…,ajn, j=1,…,m, such that (14)Hj=x∈P:aj1x1+aj2x2+⋯+ajnxn=1,Cj∩P=x∈Cj:aj1x1+aj2x2+⋯+ajnxn≤1sinceCj=posHj.From this, it follows (15)hPx=inf⁡r>0:xr∈P=inf⁡r>0:xr∈⋃j=1mCj∩P=inf⁡r>0:xr∈x∈Rn:x∈C1,a11x1+⋯+a1nxn≤1∪⋯∪x∈Rn:x∈Cm,am1x1+⋯+amnxn≤1=inf⁡r>0:∑j=1mICjxaj1x1+aj2x2+⋯+ajnxn≤r=∑j=1mICjxaj1x1+aj2x2+⋯+ajnxn,∀x=x1,…,xnT∈Rn.Representation (b2) follows by applying HP and using that (16)P=x∈Rn:A′x≤1n=x∈Rn:A1′x≤1,A2′x≤1,…,Am′x≤1=x∈Rn:max⁡A1′x,A2′x,…,Am′x≤1.


## 3. Polyhedral Generalized Surface Content Measure

### 3.1. Specific Representations

In this section, specific representations of the star-generalized surface content measureOS are considered in case that K=P is a star-shaped polyhedron and S=S 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.
LetP⊂Rn be a star-shaped polyhedron. The polyhedral star-shaped polar coordinate transformation SPOLP:Mn→Rn, Mn=[0,∞)×Mn∗, Mn∗=[0,π)×(n-2)×[0,2π) is defined by (17)x1=rNφ1,…,φn-1cos⁡φ1x2=rNφ1,…,φn-1sin⁡φ1cos⁡φ2⋮xn-1=rNφ1,…,φn-1sin⁡φ1,…,sin⁡φn-2cos⁡φn-1xn=rNφ1,…,φn-1sin⁡φ1,…,sin⁡φn-2sin⁡φn-1,where N(φ1,…,φn-1)=hP(cos⁡φ1,sin⁡φ1cos⁡φ2,…,sin⁡φ1,…,sin⁡(φn-1)).We will later also use the notationx=x(φ,r)=(x1(r,φ),…,xn(r,φ)) if x=SPOLP(r,φ).Lemma 4.
The mapSPOLP is almost one-to-one, for x≠0, its inverse SPOLP-1 is given by (18)r=hPx1,…,xn,φi=arccos⁡xi∑j=inxj21/2,i=1,…,n-2,φn-1=arccot⁡xn-1xn,xn>0,arccot⁡xn-1xn+π,xn<0and its Jacobian is (19)JSPOLPr,φ=rn-1J∗SPOLPφ,r,φ∈MnwhereJ∗SPOLPφ1,…,φn-1=1Nφn∏i=1n-1sin⁡φin-i-1.Proof.
It holds(20)hPx1,…,xn=rNφ1,…,φn-1hPcos⁡φ1,…,sin⁡φ1,…,sin⁡φn-1︸=Nφ1,…,φn-1=rand the relations of φi,  i=1,…,n-1, are the same as those in the case of usual polar coordinates. For calculating the Jacobian, we refer to the proof of Theorem 3.1 in [25] that can be extended to the n-dimensional case.Let us denote the cdf of a random vectorZ by FZ.Theorem 5.
Let(R,Φ1,…,Φn-1)T=SPOLP-1(X) be the random polyhedral star-shaped coordinates of X, where X~Φg,P; then R and (Φ1,…,Φn-1)T are stochastically independent, FR(t)=∫0t(rn-1/I(g))g(r)dr, and with φ=(φ1,…,φn-1)(21)FΦiα=1nμP∫0π​⋯∫0π​︸i-1 times∫0α∫0π​⋯∫0π​︸n-i-2 times∫02πJ∗SPOLPφdφ,∀i=1,…,n-2, α∈[0,π) and (22)FΦn-1α=1nμP∫0π​⋯∫0π​︸n-2 times∫0αJ∗SPOLPφdφ,α∈0,2π.Proof.
Applying the described method in Remark11 in [6] for elliptically contoured distributions analogously here yields the representation of FR. Furthermore, P(Φi<α)=Φg,P(A(α)) for every i=1,…,n-1, where A(α)={(r,φ1,…,φn-1)∈[0,∞)×Mn∗:φi∈[0,α)}. Thus with φ=(φ1,…,φn-1)(23)PΦi<α=Cg,P︸1/nIgμP∫0∞rn-1grdr︸=Ig∫0π​⋯∫0π​︸i-1 times∫0α∫0π​⋯∫0π​︸n-i-2 times∫02πJ∗SPOLPφdφ.Let us denote bySPOLP∗(φ)=SPOLP(1,φ) the restriction of SPOLP to the case r=1 and, analogously, by SPOLP∗-1 the inverse of the map SPOLP∗:Mn∗→S. Now, the following representations of OS can be proved.Theorem 6.
LetP⊂Rn be a star-shaped polyhedron and B∈(S∩Bn). (a)
OS satisfies the polyhedral star-shaped polar coordinate representation (24)OSB=∫SPOLP∗-1BJ∗SPOLPφdφ.(b1)
With notation as inSHP and Lemma 2 (b1), OS satisfies the star-spherical coordinate representation (25)OSB=∑j=1m∫GB∩Cj∩P1ajndϑ.(b2)
Let additionallyvoln-1 denote the volume in Rn-1, B=⋃j=1mBj where Bj⊆(Cj∩P), int(Bj)∩int(Bi)≠∅ if i≠j, i,j∈{1,…,m}; then OS satisfies the facet-content representation (26)OSB=∑j=1mvoln-1BjajTaj,whereaj=(aj1,…,ajn)T. Note that OS is proportional to the Lebesgue measure in Rn-1 in every sector Cj∩P, j=1,…,m.(c)
Let additionallyP∩Cj, j∈{1,…,m} be n-dimensional simplizes and P∩Cj and B∩Cj, j∈{1,…,m} represented according to VP. Then there exist integers kj such that B∩Cj=⋃i=1kjconv{b(j),i1,…,b(j),in} and int(conv{b(j),i11,…,b(j),i1n})∩int(conv{b(j),i21,…,b(j),i2n})=∅, i1≠i2, where conv{b(j),i1,…,b(j),in} are (n-1)-dimensional simplizes. Then OS satisfies the simplicial representation (27)OSB=1n-1!∑j=1mVB∩Cj,where(28)VB∩Cj=∑i=1kjdet⁡bj,i1,…,bj,in,ifB∩Cj≠∅,0,otherwise.(d)
If additionallyhP denotes a norm and N(ϑ)=(∇η(ϑ),-1)T is the outer normal vector to the norm sphere S at the point (ϑT,η(ϑ))T, then, with notations as in SVP, OS satisfies the dual norm representation(29)OSA=∫GAmax⁡p1TNϑ,…,plTNϑdϑ,A∈BS+,GA=ϑ∈Rn-1:∃η=ηϑwithϑT,ηT∈A.(e)
If additionallyhP denotes an antinorm and P∈AN2 and N¯(ϑ)=(∇η(ϑ),1)T is the inner normal vector to the antinorm sphere S at the point (ϑT,η(ϑ))T, then with notations as in SHP the Minkowski functional of the antipolar set P∘ of P defined by hP∘(u)=∑i=1mICiuinf⁡{uTy:y∈S∩Ci}, u∈Rn and OS satisfies the (antipolar set) representation (30)OSB=∫GBhP∘N¯ϑdϑ.Proof.
Since(31)μsectorPB,ρ=∫sectorPB,ρdx=∫0ρ∫SPOLP∗-1Brn-1J∗SPOLPφdφ,r,(5) yields (a). Analogously, (32)μsectorPB,ρ=∫sectorPB,ρdx=∫0ρrn-1dr∑i=1m∫GB∩Ci∩PJj∗ϑdϑ,where, for every ϑ∈G(Sj), Jj∗(ϑ), j=1,…,m is the value of the Jacobian of the star-spherical coordinate transformation; see [6]. Since rHj, j=1,…,m can be represented by star-spherical coordinates as (33)rHj=r·ϑ,ηjϑT∈P:ηjϑ=1ajn1-∑i=1n-1ajiϑi,j=1,…,m,it follows by applying Lemma 1 in [6] that Jj∗(ϑ)=1/|ajn|. Thus, (5) yields (b1). Considering part (b2), it holds μ(sectorP(B,1))=∑j=1mμ(sectorP(Bj,1)), where every sectorP(Bj,1), j=1,…,m is a convex cone with base Bj and cusp in the origin. Thus μ(sectorP(Bj,1))=voln-1(Bj)/(najTaj), where 1/ajTaj denotes the Euclidean length of the height of sectorP(Bj,1). According to Remark 2 in [6], it holds OS(B)=OS(S)μ(sectorP(B,1))/μ(P). Thus(34)OSB=OSSnμP︸=1∑j=1mvoln-1BjajTaj,and (b2) is proved. Considering part (c), it holds (35)μsectorPB∩Cj,1=∑i=1kjμconv0n,bj,i1,…,bj,in,j=1,…,m,and since μ(conv({0n,bj,i1,…,bj,in}))=(1/n!)det⁡bj,i1,…,bj,in it follows (36)OSB=OSSμP︸=nμsectorPB,1=1n-1!∑j=1mVB∩Cj.Considering part (d) and applying (8), it remains to prove that hP∗(N(ϑ))=max⁡{p1TN(ϑ),…,plTN(ϑ)}. According to [26] it is well known that hP∗ equals the support function of P; thus hP∗(u)=sup⁡{uTy:y∈P}, u∈Rn, and it follows hP∗(u)=max⁡{p1Tu,…,plTu}.
Considering part (e) we can use the results from Lemma1 in [6], according to which it holds (37)OSB=∫GBϑT,ηϑN¯ϑdϑ.With notations as in SHP, it follows for every i=1,…,m that S∩Ci={x∈Rn:ai1x1+⋯+ainxn=1},η(ϑ)=1/ain1-∑j=1n-1aijϑj and N¯(ϑ)=(ai1/ain,…,ai(n-1)/ain,1)T. Thus, for every (ϑT,η(ϑ))T∈S∩Ci, (38)ϑT,ηϑN¯ϑ=1ain=inf⁡N¯ϑy:y∈S∩Ci.Since the antisupport function hPF of P with respect to F is defined by
h P F ( u ) = ∑ i = 1 m I C i u inf ⁡ { u T y : y ∈ S ∩ C i } ,   u ∈ R n and it is shown in Lemma 10 in [7] that hPF=hP∘, part (e) is proved.According to [7] it is possible to represent the antipolar set P∘ of a P∈AN2 by (39)P∘=λuu:0≤λuhP∘u≤1,u∈SEn-1,where SE(n-1) denotes the Euclidean unit sphere in Rn. For an illustration, we refer to Section 4.2, where antipolar sets of w-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 numberπ have been discussed to a certain extent first in [27] for l2,p-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 ln,p-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(40)μPr=∫0rOSSρdρreflects a certain generalization of the method of indivisibles of Cavalieri and Torricelli and that the ratios (41)μPrrn=OSSrnrn-1not only do coincide but are even independent of r. This motivates the following definition.Definition 7.
The polyhedral star ball numberπP is defined by πP=μ(P).
Note thatπP=OS(S)/n. For concrete values of πP 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, πP 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 onS as(42)ωSA=OSAOSS,A∈BS.Note 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 ωS can be found in Example A.1 in the Appendix.Let(Ω,A,P) be a probability space and G a random vector being defined on Ω and taking values in Rn. Assume further that G follows the uniform probability distribution on P; that is, P(G∈A)=μ(A)/μ(P), for A∈(P∩Bn), and put Y=G/hP(G), 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 vectorY follows the polyhedral star-shaped generalized uniform distribution on S.


## 3.1. Specific Representations

In this section, specific representations of the star-generalized surface content measureOS are considered in case that K=P is a star-shaped polyhedron and S=S 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.
LetP⊂Rn be a star-shaped polyhedron. The polyhedral star-shaped polar coordinate transformation SPOLP:Mn→Rn, Mn=[0,∞)×Mn∗, Mn∗=[0,π)×(n-2)×[0,2π) is defined by (17)x1=rNφ1,…,φn-1cos⁡φ1x2=rNφ1,…,φn-1sin⁡φ1cos⁡φ2⋮xn-1=rNφ1,…,φn-1sin⁡φ1,…,sin⁡φn-2cos⁡φn-1xn=rNφ1,…,φn-1sin⁡φ1,…,sin⁡φn-2sin⁡φn-1,where N(φ1,…,φn-1)=hP(cos⁡φ1,sin⁡φ1cos⁡φ2,…,sin⁡φ1,…,sin⁡(φn-1)).We will later also use the notationx=x(φ,r)=(x1(r,φ),…,xn(r,φ)) if x=SPOLP(r,φ).Lemma 4.
The mapSPOLP is almost one-to-one, for x≠0, its inverse SPOLP-1 is given by (18)r=hPx1,…,xn,φi=arccos⁡xi∑j=inxj21/2,i=1,…,n-2,φn-1=arccot⁡xn-1xn,xn>0,arccot⁡xn-1xn+π,xn<0and its Jacobian is (19)JSPOLPr,φ=rn-1J∗SPOLPφ,r,φ∈MnwhereJ∗SPOLPφ1,…,φn-1=1Nφn∏i=1n-1sin⁡φin-i-1.Proof.
It holds(20)hPx1,…,xn=rNφ1,…,φn-1hPcos⁡φ1,…,sin⁡φ1,…,sin⁡φn-1︸=Nφ1,…,φn-1=rand the relations of φi,  i=1,…,n-1, are the same as those in the case of usual polar coordinates. For calculating the Jacobian, we refer to the proof of Theorem 3.1 in [25] that can be extended to the n-dimensional case.Let us denote the cdf of a random vectorZ by FZ.Theorem 5.
Let(R,Φ1,…,Φn-1)T=SPOLP-1(X) be the random polyhedral star-shaped coordinates of X, where X~Φg,P; then R and (Φ1,…,Φn-1)T are stochastically independent, FR(t)=∫0t(rn-1/I(g))g(r)dr, and with φ=(φ1,…,φn-1)(21)FΦiα=1nμP∫0π​⋯∫0π​︸i-1 times∫0α∫0π​⋯∫0π​︸n-i-2 times∫02πJ∗SPOLPφdφ,∀i=1,…,n-2, α∈[0,π) and (22)FΦn-1α=1nμP∫0π​⋯∫0π​︸n-2 times∫0αJ∗SPOLPφdφ,α∈0,2π.Proof.
Applying the described method in Remark11 in [6] for elliptically contoured distributions analogously here yields the representation of FR. Furthermore, P(Φi<α)=Φg,P(A(α)) for every i=1,…,n-1, where A(α)={(r,φ1,…,φn-1)∈[0,∞)×Mn∗:φi∈[0,α)}. Thus with φ=(φ1,…,φn-1)(23)PΦi<α=Cg,P︸1/nIgμP∫0∞rn-1grdr︸=Ig∫0π​⋯∫0π​︸i-1 times∫0α∫0π​⋯∫0π​︸n-i-2 times∫02πJ∗SPOLPφdφ.Let us denote bySPOLP∗(φ)=SPOLP(1,φ) the restriction of SPOLP to the case r=1 and, analogously, by SPOLP∗-1 the inverse of the map SPOLP∗:Mn∗→S. Now, the following representations of OS can be proved.Theorem 6.
LetP⊂Rn be a star-shaped polyhedron and B∈(S∩Bn). (a)
OS satisfies the polyhedral star-shaped polar coordinate representation (24)OSB=∫SPOLP∗-1BJ∗SPOLPφdφ.(b1)
With notation as inSHP and Lemma 2 (b1), OS satisfies the star-spherical coordinate representation (25)OSB=∑j=1m∫GB∩Cj∩P1ajndϑ.(b2)
Let additionallyvoln-1 denote the volume in Rn-1, B=⋃j=1mBj where Bj⊆(Cj∩P), int(Bj)∩int(Bi)≠∅ if i≠j, i,j∈{1,…,m}; then OS satisfies the facet-content representation (26)OSB=∑j=1mvoln-1BjajTaj,whereaj=(aj1,…,ajn)T. Note that OS is proportional to the Lebesgue measure in Rn-1 in every sector Cj∩P, j=1,…,m.(c)
Let additionallyP∩Cj, j∈{1,…,m} be n-dimensional simplizes and P∩Cj and B∩Cj, j∈{1,…,m} represented according to VP. Then there exist integers kj such that B∩Cj=⋃i=1kjconv{b(j),i1,…,b(j),in} and int(conv{b(j),i11,…,b(j),i1n})∩int(conv{b(j),i21,…,b(j),i2n})=∅, i1≠i2, where conv{b(j),i1,…,b(j),in} are (n-1)-dimensional simplizes. Then OS satisfies the simplicial representation (27)OSB=1n-1!∑j=1mVB∩Cj,where(28)VB∩Cj=∑i=1kjdet⁡bj,i1,…,bj,in,ifB∩Cj≠∅,0,otherwise.(d)
If additionallyhP denotes a norm and N(ϑ)=(∇η(ϑ),-1)T is the outer normal vector to the norm sphere S at the point (ϑT,η(ϑ))T, then, with notations as in SVP, OS satisfies the dual norm representation(29)OSA=∫GAmax⁡p1TNϑ,…,plTNϑdϑ,A∈BS+,GA=ϑ∈Rn-1:∃η=ηϑwithϑT,ηT∈A.(e)
If additionallyhP denotes an antinorm and P∈AN2 and N¯(ϑ)=(∇η(ϑ),1)T is the inner normal vector to the antinorm sphere S at the point (ϑT,η(ϑ))T, then with notations as in SHP the Minkowski functional of the antipolar set P∘ of P defined by hP∘(u)=∑i=1mICiuinf⁡{uTy:y∈S∩Ci}, u∈Rn and OS satisfies the (antipolar set) representation (30)OSB=∫GBhP∘N¯ϑdϑ.Proof.
Since(31)μsectorPB,ρ=∫sectorPB,ρdx=∫0ρ∫SPOLP∗-1Brn-1J∗SPOLPφdφ,r,(5) yields (a). Analogously, (32)μsectorPB,ρ=∫sectorPB,ρdx=∫0ρrn-1dr∑i=1m∫GB∩Ci∩PJj∗ϑdϑ,where, for every ϑ∈G(Sj), Jj∗(ϑ), j=1,…,m is the value of the Jacobian of the star-spherical coordinate transformation; see [6]. Since rHj, j=1,…,m can be represented by star-spherical coordinates as (33)rHj=r·ϑ,ηjϑT∈P:ηjϑ=1ajn1-∑i=1n-1ajiϑi,j=1,…,m,it follows by applying Lemma 1 in [6] that Jj∗(ϑ)=1/|ajn|. Thus, (5) yields (b1). Considering part (b2), it holds μ(sectorP(B,1))=∑j=1mμ(sectorP(Bj,1)), where every sectorP(Bj,1), j=1,…,m is a convex cone with base Bj and cusp in the origin. Thus μ(sectorP(Bj,1))=voln-1(Bj)/(najTaj), where 1/ajTaj denotes the Euclidean length of the height of sectorP(Bj,1). According to Remark 2 in [6], it holds OS(B)=OS(S)μ(sectorP(B,1))/μ(P). Thus(34)OSB=OSSnμP︸=1∑j=1mvoln-1BjajTaj,and (b2) is proved. Considering part (c), it holds (35)μsectorPB∩Cj,1=∑i=1kjμconv0n,bj,i1,…,bj,in,j=1,…,m,and since μ(conv({0n,bj,i1,…,bj,in}))=(1/n!)det⁡bj,i1,…,bj,in it follows (36)OSB=OSSμP︸=nμsectorPB,1=1n-1!∑j=1mVB∩Cj.Considering part (d) and applying (8), it remains to prove that hP∗(N(ϑ))=max⁡{p1TN(ϑ),…,plTN(ϑ)}. According to [26] it is well known that hP∗ equals the support function of P; thus hP∗(u)=sup⁡{uTy:y∈P}, u∈Rn, and it follows hP∗(u)=max⁡{p1Tu,…,plTu}.
Considering part (e) we can use the results from Lemma1 in [6], according to which it holds (37)OSB=∫GBϑT,ηϑN¯ϑdϑ.With notations as in SHP, it follows for every i=1,…,m that S∩Ci={x∈Rn:ai1x1+⋯+ainxn=1},η(ϑ)=1/ain1-∑j=1n-1aijϑj and N¯(ϑ)=(ai1/ain,…,ai(n-1)/ain,1)T. Thus, for every (ϑT,η(ϑ))T∈S∩Ci, (38)ϑT,ηϑN¯ϑ=1ain=inf⁡N¯ϑy:y∈S∩Ci.Since the antisupport function hPF of P with respect to F is defined by
h P F ( u ) = ∑ i = 1 m I C i u inf ⁡ { u T y : y ∈ S ∩ C i } ,   u ∈ R n and it is shown in Lemma 10 in [7] that hPF=hP∘, part (e) is proved.According to [7] it is possible to represent the antipolar set P∘ of a P∈AN2 by (39)P∘=λuu:0≤λuhP∘u≤1,u∈SEn-1,where SE(n-1) denotes the Euclidean unit sphere in Rn. For an illustration, we refer to Section 4.2, where antipolar sets of w-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 numberπ have been discussed to a certain extent first in [27] for l2,p-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 ln,p-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(40)μPr=∫0rOSSρdρreflects a certain generalization of the method of indivisibles of Cavalieri and Torricelli and that the ratios (41)μPrrn=OSSrnrn-1not only do coincide but are even independent of r. This motivates the following definition.Definition 7.
The polyhedral star ball numberπP is defined by πP=μ(P).
Note thatπP=OS(S)/n. For concrete values of πP 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, πP 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 onS as(42)ωSA=OSAOSS,A∈BS.Note 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 ωS can be found in Example A.1 in the Appendix.Let(Ω,A,P) be a probability space and G a random vector being defined on Ω and taking values in Rn. Assume further that G follows the uniform probability distribution on P; that is, P(G∈A)=μ(A)/μ(P), for A∈(P∩Bn), and put Y=G/hP(G), 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 vectorY follows the polyhedral star-shaped generalized uniform distribution on S.


## 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 [6–8, 12, 20, 21, 28], where stochastic representations like (7) were repeatedly exploited.A random vectorX:Ω→Rn is said to follow a polyhedral star-shaped distribution if there exists a random variable R:Ω→[0,∞) such that X satisfies the stochastic representation(43)X=dR·US,where US~ωS is polyhedral star-shaped generalized uniformly distributed on S and US and R are stochastically independent. The set of all polyhedral star-shaped distributions on Bn will be denoted by PStSh(n) and its subset of continuous distributions by (44)CPStShn=Φg,P:Pis a star-shaped polyhedron,gis a dgf.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 [28–30].Proposition 9.
Let the assumptions from Theorem6 be fulfilled and B∈Bn. Then Φg,P(B) allows correspondingly (a)
polyhedral star-shaped polar coordinate representation(45)Φg,PB=1nπPIg∫0∞∫SPOLP∗-11/rB∩Srn-1grJ∗SPOLPφdφdr,(b1)
star-spherical coordinate representation(46)Φg,PB=1nπPIg∫0∞rn-1gr∑j=1m∫G1/rB∩S∩Cj∩P1ajndϑdr,(b2)
facet-content representation(47)Φg,PB=1nπPIg∫0∞rn-1gr∑j=1mvoln-1BjrajTajdr,if additionally1/rB∩S=⋃j=1mBj(r) where Bj(r)⊆(Cj∩P) and int(Bi(r))∩int(Bj(r))=∅, if i≠j,i,j∈{1,…,m},(c)
simplicial representation(48)Φg,PB=1nπPIgn-1!∫0∞rn-1gr∑j=1mV1rB∩S∩Cjdr,(d)
dual norm representation(49)Φg,PB=1nπPIg∫0∞∫G1/rB∩Srn-1grmax⁡p1TNϑ,…,plTNϑdϑdr,(e)
(antipolar set) representation(50)Φg,PB=1nπPIg∫0∞∫G1/rB∩Srn-1grhP∘N¯ϑdϑdr,(f)
sector measure representation(51)Φg,PB=1πPIg∫0∞rn-1grμsectorP1rB∩S,1dr.Note that for the sector measure also the notion of cone measure is used; see [17, 31]Finally note that ifX~Φg,P then we write (43) as(52)X=dRg·US,where the nonnegative random variable Rg is independent of US.


### 4.2. The Class ofw-Star Contoured Distributions in R2

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. LetPw⊂R2 be a w (vertices) driven star-shaped polygon defined for arbitrary w∈(0,∞) by the vertices q1=(1,0), q2=(w,w), q3=(0,1), q4=(-w,w), q5=(-1,0), q6=(-w,-w), q7=(0,-1), and q8=(w,-w). We call Pw a w-star. For an illustration of different w-stars, see Figure 1. Note that a w-star appears to be convex if w∈[1/2,1] and radially concave if w∈(0,1/2)∪(1,∞). For representing the Minkowski functional hPw of a w-star, it is convenient to make use of the fans C1=⋃i=14C1,i and C2=⋃i=14C2,i if w∈(0,1) or w≥1, respectively; see again Figure 1. If w∈(0,1) then(53)hPwx,y=IC1,1x,yx+1-wwy+IC1,2x,y1-wwx+y+IC1,3x,y-x+1-wwy+IC1,4x,y1-wwx-yand if w≥1 then (54)hPwx,y=IC2,1x,ymax⁡x,y+1-wwmin⁡x,y+IC2,2x,ymax⁡-x,y+1-wwmin⁡-x,y+IC2,3x,ymax⁡-x,-y+1-wwmin⁡-x,-y+IC2,4x,ymax⁡x,-y+1-wwmin⁡x,-y.Surprisingly enough, it follows that hPw(x,y)=max{|x|,|y|}+(1-w)/wmin{|x|,|y|} for every w∈(0,∞). This result enables us to introduce the w-star contoured densities (55)φg,Pwx,y=18wIggmax⁡x,y+1-wwmin⁡x,y,x,yT∈R2.Note that since μ(Pw)=4w the normalizing constant applies by (4). If w∈[1/2,1] then hPw denotes a norm. Thus, applying Theorem 6 (d), we can represent OSw of a set A∈BSw+ by the dual norm representation as (56)OSwA=∫GAmax⁡N1ϑ,N2ϑ,wN1ϑ+N2ϑdϑ,where N(ϑ)=(N1(ϑ),N2(ϑ))T. 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
w-stars in case 0<w<1 (a) and in case w≥1 (b).
(a) (b)Ifw∈(0,1/2)∪(1,∞) then hPw denotes an antinorm. Thus, applying Theorem 6 (e), we can represent OSw of a set B∈BSw by the (antipolar set) representation (57)OSwB=∫GB∑i=18ICiai1ai2,1Twai1ai2+1dϑ=∫GBwN¯1ϑ+N¯2ϑdϑ,where the sectors Ci, i=1,…,8, are equivalent to those in Example A.2, ai1 and ai2, i=1,…,8, are equivalent to the entries of the matrix A in Example A.2, N¯(ϑ)=(N¯1(ϑ),N¯2(ϑ))T, and the antipolar set Pw∘ of Pw is the l1-ball of Minkowski radius 1/w,Pw∘={(x1,x2)T∈R2:x1+x2≤1/w}.


## 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 [6–8, 12, 20, 21, 28], where stochastic representations like (7) were repeatedly exploited.A random vectorX:Ω→Rn is said to follow a polyhedral star-shaped distribution if there exists a random variable R:Ω→[0,∞) such that X satisfies the stochastic representation(43)X=dR·US,where US~ωS is polyhedral star-shaped generalized uniformly distributed on S and US and R are stochastically independent. The set of all polyhedral star-shaped distributions on Bn will be denoted by PStSh(n) and its subset of continuous distributions by (44)CPStShn=Φg,P:Pis a star-shaped polyhedron,gis a dgf.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 [28–30].Proposition 9.
Let the assumptions from Theorem6 be fulfilled and B∈Bn. Then Φg,P(B) allows correspondingly (a)
polyhedral star-shaped polar coordinate representation(45)Φg,PB=1nπPIg∫0∞∫SPOLP∗-11/rB∩Srn-1grJ∗SPOLPφdφdr,(b1)
star-spherical coordinate representation(46)Φg,PB=1nπPIg∫0∞rn-1gr∑j=1m∫G1/rB∩S∩Cj∩P1ajndϑdr,(b2)
facet-content representation(47)Φg,PB=1nπPIg∫0∞rn-1gr∑j=1mvoln-1BjrajTajdr,if additionally1/rB∩S=⋃j=1mBj(r) where Bj(r)⊆(Cj∩P) and int(Bi(r))∩int(Bj(r))=∅, if i≠j,i,j∈{1,…,m},(c)
simplicial representation(48)Φg,PB=1nπPIgn-1!∫0∞rn-1gr∑j=1mV1rB∩S∩Cjdr,(d)
dual norm representation(49)Φg,PB=1nπPIg∫0∞∫G1/rB∩Srn-1grmax⁡p1TNϑ,…,plTNϑdϑdr,(e)
(antipolar set) representation(50)Φg,PB=1nπPIg∫0∞∫G1/rB∩Srn-1grhP∘N¯ϑdϑdr,(f)
sector measure representation(51)Φg,PB=1πPIg∫0∞rn-1grμsectorP1rB∩S,1dr.Note that for the sector measure also the notion of cone measure is used; see [17, 31]Finally note that ifX~Φg,P then we write (43) as(52)X=dRg·US,where the nonnegative random variable Rg is independent of US.


## 4.2. The Class ofw-Star Contoured Distributions in R2

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. LetPw⊂R2 be a w (vertices) driven star-shaped polygon defined for arbitrary w∈(0,∞) by the vertices q1=(1,0), q2=(w,w), q3=(0,1), q4=(-w,w), q5=(-1,0), q6=(-w,-w), q7=(0,-1), and q8=(w,-w). We call Pw a w-star. For an illustration of different w-stars, see Figure 1. Note that a w-star appears to be convex if w∈[1/2,1] and radially concave if w∈(0,1/2)∪(1,∞). For representing the Minkowski functional hPw of a w-star, it is convenient to make use of the fans C1=⋃i=14C1,i and C2=⋃i=14C2,i if w∈(0,1) or w≥1, respectively; see again Figure 1. If w∈(0,1) then(53)hPwx,y=IC1,1x,yx+1-wwy+IC1,2x,y1-wwx+y+IC1,3x,y-x+1-wwy+IC1,4x,y1-wwx-yand if w≥1 then (54)hPwx,y=IC2,1x,ymax⁡x,y+1-wwmin⁡x,y+IC2,2x,ymax⁡-x,y+1-wwmin⁡-x,y+IC2,3x,ymax⁡-x,-y+1-wwmin⁡-x,-y+IC2,4x,ymax⁡x,-y+1-wwmin⁡x,-y.Surprisingly enough, it follows that hPw(x,y)=max{|x|,|y|}+(1-w)/wmin{|x|,|y|} for every w∈(0,∞). This result enables us to introduce the w-star contoured densities (55)φg,Pwx,y=18wIggmax⁡x,y+1-wwmin⁡x,y,x,yT∈R2.Note that since μ(Pw)=4w the normalizing constant applies by (4). If w∈[1/2,1] then hPw denotes a norm. Thus, applying Theorem 6 (d), we can represent OSw of a set A∈BSw+ by the dual norm representation as (56)OSwA=∫GAmax⁡N1ϑ,N2ϑ,wN1ϑ+N2ϑdϑ,where N(ϑ)=(N1(ϑ),N2(ϑ))T. 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
w-stars in case 0<w<1 (a) and in case w≥1 (b).
(a) (b)Ifw∈(0,1/2)∪(1,∞) then hPw denotes an antinorm. Thus, applying Theorem 6 (e), we can represent OSw of a set B∈BSw by the (antipolar set) representation (57)OSwB=∫GB∑i=18ICiai1ai2,1Twai1ai2+1dϑ=∫GBwN¯1ϑ+N¯2ϑdϑ,where the sectors Ci, i=1,…,8, are equivalent to those in Example A.2, ai1 and ai2, i=1,…,8, are equivalent to the entries of the matrix A in Example A.2, N¯(ϑ)=(N¯1(ϑ),N¯2(ϑ))T, and the antipolar set Pw∘ of Pw is the l1-ball of Minkowski radius 1/w,Pw∘={(x1,x2)T∈R2:x1+x2≤1/w}.


## 5. Expectations

### 5.1. Moments of Polyhedral Star-Shaped Distributions

LetG=(G1,…,Gn)T be uniformly distributed on a star-shaped polyhedron P. Such vector has dgf g(r)=I[0,1](r). Since I(g)=1/n the density of G is(58)φg,Px=I0,1hPxμP=IPxμP,x∈Rn.With E(Gik)=1/μ(P)∫Pxikdμ, i=1,…,n, the vector of moments of order k is defined as E(Gk)=(E(G1k),…,E(Gnk))T. Mixed moments of order k=k1+⋯+kn,kj≥0,j∈{1,…,n} are (59)EG1k1·…·Gnkn=1μP∫Px1k1·…·xkkndμ.The numerical computation of these moments of G 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 VP or HP, and the results are exact if the entries (vertices in case of VP or A and b in case of HP) are rational numbers; see [32–34]. Note that star-shaped polyhedra can always be divided into disjoint convex polyhedra P∩Cj,j∈{1,…,m}, say. Moreover, μ(P) can also be computed by integrating the function f(x)=1 over P.Theorem 10.
LetP⊂Rn be a star-shaped polyhedron, G=(G1,…,Gn)T uniformly distributed on P and X=(X1,…,Xn)T~Φg,P. (a)
If the momentE(Rgk) exists then E(Xk)=n+k/nE(Rgk)E(Gk).(b)
IfE(Rgk) exists then, for all kj≥0, j∈{1,…,n} with k=k1+⋯+kn, (60)EX1k1·…·Xnkn=n+knERgkEG1k1·…·Gnkn.(c)
In case of existence, the covariance matrixcov(X) of X allows the representation (61)covX=n+2nERg2EGGT-n+1n2ERg2EGEGT.Proof.
By the stochastic representation in (52), we have E(Xk)=E(Rgk·USk)=E(Rgk)E(USk). According to Theorem 8, G=dhP(G)·US where hP(G) and US are stochastically independent, hP(G) follows the density function f(hP(G))(t)=I[0,1](t)ntn-1, and(62)EhPGk=∫-∞∞tkfhPGtdt=n∫01tn+k-1dt=nn+k.Since E(Gk)=E((hP(G))k)E(USk) it follows (a). Parts (b) and (c) can be proved analogously.For basic types of dgfg and representations of the moments of Rg, 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 ln,1-symmetrically distributed random vectors in [36] and for ln,p-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.Figure 2
ϕ g 0 , w with w=0.25 (a) and its level sets (b).
(a) (b)Figure 3
ϕ g 0 , w with w=0.8 (a) and its level sets (b).
(a) (b)Theorem 11.
IfX~Φg,P then its characteristic function allows the representation(63)ϕXt=Cg,Pn∫0∞∫Mn∗cos⁡∑i=1ntixir,φJ∗SPOLPφdφrn-1grdr+i∫0∞∫Mn∗sin⁡∑i=1ntixir,φJ∗SPOLPφdφrn-1grdr,t∈Rn.Proof.
Applying the stochastic representation (52) and Theorem 1.1.6 in [38], it is proved in Theorem 9 in [6] that(64)ϕXt=1Ig∫0∞ϕUSrtrn-1grdr,t∈Rn.Applying the representation of ϕUS(t) in Remark 10 (a) of [6] and the polyhedral star-shaped polar coordinate transformation to (64) yields (63).
Note that ifUS is symmetrically distributed with respect to the origin, US=d-US, the imaginary part in equation (63) vanishes. To be specific, let (X,Y)T follow the w-star contoured distribution Φg,Pw. The characteristic function of (X,Y)T is (65)ϕg,wt1,t2=18wIg·∫0∞∫02πcos⁡t1rcosφ+t2rsinφmax⁡cos⁡φ,sin⁡φ+1-w/wmin⁡cos⁡φ,sin⁡φ·rgrmax⁡cos⁡φ,sin⁡φ+1-w/wmin⁡cos⁡φ,sin⁡φ2dφdr.Note that the use of polyhedral star-shaped polar coordinates yields an integral representation of the characteristic function where the range of integration is much easier to describe than if one makes use of other types of coordinates. We exploit this fact for drawing Figures 2 and 3 where this characteristic function is visualized for the particular dgf g0(r)=(1+r2)-2.


### 5.3. Moment Generating Functions

Considering a random vectorX~Φg,P the moment generating function MX:Rn→R can be formally represented by(66)MXt1,…,tn=Cg,P∫Rnet1x1+⋯+tnxnghPx1,…,xndx1,…,xn.It is possible to specify representation (66) given a certain type of dgf and a certain representation of P. The results are shown in the following theorem.Theorem 12.
LetX~Φg,P and the special Kotz-type dgf g(r)=e-βr, β>0 be given. If hP is represented by Lemma 2 (b1) and it exists an ϵ>0 such that (67)hPtx=∑i=1mICixai1-t1βx1+⋯+ain-tnβxnfor all t=(t1,…,tn)T∈[-ϵ,ϵ]×n is the Minkowski functional of the star-shaped polyhedron P(t)={x∈Rn:hP(t)(x)≤1} then(68)MXt=πPtπP,t=t1,…,tnT∈-ϵ,ϵ×n.Proof.
SinceC(g,P)=βn/nπPΓn it holds (69)MXt=βnnπPΓn∫Rnet,xe-βhPxdx=βnnπPΓn∑i=1m∫Cie-βai1-t1/βx1+⋯+ain-tn/βxndx1,…,xn=βnnπPΓn∫Rne-βhPtxdx︸=nπPtΓn/βn=πPtπP.The specific representation (68) of the moment generating function can be applied to compute vectors of moments of polyhedral star-shaped distributions. For an illustration, we refer to Example A.2 in the Appendix.


## 5.1. Moments of Polyhedral Star-Shaped Distributions

LetG=(G1,…,Gn)T be uniformly distributed on a star-shaped polyhedron P. Such vector has dgf g(r)=I[0,1](r). Since I(g)=1/n the density of G is(58)φg,Px=I0,1hPxμP=IPxμP,x∈Rn.With E(Gik)=1/μ(P)∫Pxikdμ, i=1,…,n, the vector of moments of order k is defined as E(Gk)=(E(G1k),…,E(Gnk))T. Mixed moments of order k=k1+⋯+kn,kj≥0,j∈{1,…,n} are (59)EG1k1·…·Gnkn=1μP∫Px1k1·…·xkkndμ.The numerical computation of these moments of G 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 VP or HP, and the results are exact if the entries (vertices in case of VP or A and b in case of HP) are rational numbers; see [32–34]. Note that star-shaped polyhedra can always be divided into disjoint convex polyhedra P∩Cj,j∈{1,…,m}, say. Moreover, μ(P) can also be computed by integrating the function f(x)=1 over P.Theorem 10.
LetP⊂Rn be a star-shaped polyhedron, G=(G1,…,Gn)T uniformly distributed on P and X=(X1,…,Xn)T~Φg,P. (a)
If the momentE(Rgk) exists then E(Xk)=n+k/nE(Rgk)E(Gk).(b)
IfE(Rgk) exists then, for all kj≥0, j∈{1,…,n} with k=k1+⋯+kn, (60)EX1k1·…·Xnkn=n+knERgkEG1k1·…·Gnkn.(c)
In case of existence, the covariance matrixcov(X) of X allows the representation (61)covX=n+2nERg2EGGT-n+1n2ERg2EGEGT.Proof.
By the stochastic representation in (52), we have E(Xk)=E(Rgk·USk)=E(Rgk)E(USk). According to Theorem 8, G=dhP(G)·US where hP(G) and US are stochastically independent, hP(G) follows the density function f(hP(G))(t)=I[0,1](t)ntn-1, and(62)EhPGk=∫-∞∞tkfhPGtdt=n∫01tn+k-1dt=nn+k.Since E(Gk)=E((hP(G))k)E(USk) it follows (a). Parts (b) and (c) can be proved analogously.For basic types of dgfg and representations of the moments of Rg, 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 ln,1-symmetrically distributed random vectors in [36] and for ln,p-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.Figure 2
ϕ g 0 , w with w=0.25 (a) and its level sets (b).
(a) (b)Figure 3
ϕ g 0 , w with w=0.8 (a) and its level sets (b).
(a) (b)Theorem 11.
IfX~Φg,P then its characteristic function allows the representation(63)ϕXt=Cg,Pn∫0∞∫Mn∗cos⁡∑i=1ntixir,φJ∗SPOLPφdφrn-1grdr+i∫0∞∫Mn∗sin⁡∑i=1ntixir,φJ∗SPOLPφdφrn-1grdr,t∈Rn.Proof.
Applying the stochastic representation (52) and Theorem 1.1.6 in [38], it is proved in Theorem 9 in [6] that(64)ϕXt=1Ig∫0∞ϕUSrtrn-1grdr,t∈Rn.Applying the representation of ϕUS(t) in Remark 10 (a) of [6] and the polyhedral star-shaped polar coordinate transformation to (64) yields (63).
Note that ifUS is symmetrically distributed with respect to the origin, US=d-US, the imaginary part in equation (63) vanishes. To be specific, let (X,Y)T follow the w-star contoured distribution Φg,Pw. The characteristic function of (X,Y)T is (65)ϕg,wt1,t2=18wIg·∫0∞∫02πcos⁡t1rcosφ+t2rsinφmax⁡cos⁡φ,sin⁡φ+1-w/wmin⁡cos⁡φ,sin⁡φ·rgrmax⁡cos⁡φ,sin⁡φ+1-w/wmin⁡cos⁡φ,sin⁡φ2dφdr.Note that the use of polyhedral star-shaped polar coordinates yields an integral representation of the characteristic function where the range of integration is much easier to describe than if one makes use of other types of coordinates. We exploit this fact for drawing Figures 2 and 3 where this characteristic function is visualized for the particular dgf g0(r)=(1+r2)-2.


## 5.3. Moment Generating Functions

Considering a random vectorX~Φg,P the moment generating function MX:Rn→R can be formally represented by(66)MXt1,…,tn=Cg,P∫Rnet1x1+⋯+tnxnghPx1,…,xndx1,…,xn.It is possible to specify representation (66) given a certain type of dgf and a certain representation of P. The results are shown in the following theorem.Theorem 12.
LetX~Φg,P and the special Kotz-type dgf g(r)=e-βr, β>0 be given. If hP is represented by Lemma 2 (b1) and it exists an ϵ>0 such that (67)hPtx=∑i=1mICixai1-t1βx1+⋯+ain-tnβxnfor all t=(t1,…,tn)T∈[-ϵ,ϵ]×n is the Minkowski functional of the star-shaped polyhedron P(t)={x∈Rn:hP(t)(x)≤1} then(68)MXt=πPtπP,t=t1,…,tnT∈-ϵ,ϵ×n.Proof.
SinceC(g,P)=βn/nπPΓn it holds (69)MXt=βnnπPΓn∫Rnet,xe-βhPxdx=βnnπPΓn∑i=1m∫Cie-βai1-t1/βx1+⋯+ain-tn/βxndx1,…,xn=βnnπPΓn∫Rne-βhPtxdx︸=nπPtΓn/βn=πPtπP.The specific representation (68) of the moment generating function can be applied to compute vectors of moments of polyhedral star-shaped distributions. For an illustration, we refer to Example A.2 in the Appendix.


## 6. Simulation of Polyhedral Star-Shaped Distributions

The stochastic representations of Theorem8 and (52) offer the possibility to generate polyhedral star-shaped distributed random points for a given contour defining star-shaped polyhedron P⊂Rn and a dgf g. By generating a random number Rg from the density fRg(r)=I(g)-1rn-1g(r) and independently of this a polyhedral star-shaped generalized uniformly distributed random point US on S, the product RgUS gives the desired polyhedral star-shaped distributed random point. US can be simulated by generating a uniformly on P distributed random point G and applying Theorem 8; that is, US=G/hP(G). It remains the simulation of G. To this end, we use a cuboid Q⊂Rn satisfying P⊆Q and apply the rejection method. For basic details on rejection sampling, we refer to [39]. Applications of the rejection method to the simulation of the p-generalized Gaussian distribution can be found in [29] and to the simulation of the uniform distribution on platonic bodies elsewhere. All single steps are summarized in Algorithm 1.Algorithm 1:Sampling algorithm for polyhedral star-shaped distributions Φg,P.
Input: g,hP,Q=[qmin,1,qmax,1]×⋯×[qmin,n,qmax,n] Output: X from Φg,P Algorithm (1) Sample Rg from fRg(r)=I(g)-1rn-1g(r). (2) Sample G uniformly from Q until hP(G)≤1. (3) Build US=G/hP(G) componentwise. (4) Build X=Rg⋅US componentwise. Return  XThe sampling method in step (1) has to be specified in accordance with the chosen dgfg. For the dgfs in Table 1, sampling methods can be found in [12]. Proving that G is uniformly distributed on P and the stopping time of step (2) is finite can be done analogously to Appendix 1 of [29]. A suitable cuboid Q can be found as follows. If P is represented by SVP, one can put(70)qmin,i=min⁡pj,ki,j=1,…,m,k=1,…,lji=1,…,n,qmax,i=max⁡pj,ki,j=1,…,m,k=1,…,lji=1,…,n,where pj,k[i] denotes the ith component of the vertex pj,k. If P is represented by SHP, one can solve for every i=1,…,n, the optimization problems: minimize qmin,i=xi (maximize qmax,i=xi) subject to hP(x1,…,xn)≤1. Since it appears to be easier to find a suitable Q in case of SVP, note that it is possible to transform P from SHP to SVP according to [40, 41].Example 13.
LetX be w-star contoured distributed, X~Φg,Pw with the Kotz-type dgf g(r)=gK(M,β,γ)(r)=r2M-2e-βr2γ where M,β,γ∈(0,∞). Figures 4, 5, and 6 show 2000 points simulated independently according to Φg,Pw.Figure 4
2000 points (a) simulated according toφgK(1,0.5,1),P0.3 (b).
(a) (b)Figure 5
2000 points (a) simulated according toφgK(3,0.5,1),P0.8 (b).
(a) (b)Figure 6
2000 points (a) simulated according toφgK(1,5,2),P3 (b).
(a) (b)


## 7. Location-Scale Transformations of Continuous Polyhedral Star-Shaped Distributions

Throughout this section, we assume thatRg has a finite second-order moment and use the notation X~Φg,P,ϑ,Ψ, where ϑ and Ψ denote expectation vector and covariance matrix of X, respectively. We consider location and scale transformations of X and study transformed density level sets.Theorem 14.
LetX~Φg,P,ϑ,Ψ with symmetric and positive definite matrix Ψ. If Σ∈Rn×n is another symmetric and positive definite matrix and μ∈Rn then there exist regular lower triangular matrices D and V from Rn×n such that DDT=Ψ, VVT=Σ and the random vector Z=VD-1(X-ϑ)+μ follows the distribution Φg,PLT,μ,Σ, where PLT=VD-1(P-ϑ)+μ, and the density is represented as (71)φg,PLTz=det⁡DCg,Pdet⁡VghPDV-1z-μ+ϑ,z∈Rn.Proof.
For the existence ofD and V, see [42]. Furthermore, E(Z)=VD-1(E(X)-ϑ)+μ=μ, and cov(Z)=VD-1Ψ(D-1)TVT=VVT=Σ. The representation of φg,P(z) follows by the density transformation formula.For an example of the application of the linear transformation in Theorem14, see Example A.3 in the Appendix. Note that the linear transformation does not only transform location and scale of the given random vector but also transforms the contour defining star-shaped polyhedron P into a polyhedron being generally not congruent with P. Referring to this, see again Figures 7 and 8. Since noncongruent change of shape is not always a desired property, because P may be strictly chosen, we introduce a second, orthogonal transformation by using Givens rotations and Givens matrices. For further information about Givens rotations and matrices, see [42].Figure 7
TetragonP (a) and level sets of the density of X (b) in Example A.3.
(a) (b)Figure 8
Linear transformed contour defining tetragonPLT (a) and density level sets of the density of Y (b) in Example A.3.
(a) (b)Definition 15.
Letl∈{1,…,n-1} and k∈{l+1,…,n}. The n-dimensional (l,k)-Givens matrix Sn(l,k;α)=Sn(l,k;α)(i,j)i,j∈{1,…,n}∈Rn×n is defined by (72)Snl,k;αi,j=cosα,i=j=l∨i=j=k,sinα,i=l∧j=k,-sinα,i=k∧j=l,1,i=j∧i≠l∧i≠k,0,otherwise.Note that, for every dimension n, there exist n(n-1)/2 Givens matrices. A Givens rotation of a vector x∈Rn defined by Sn(l,k;α)x is an orthogonal transformation and rotates x within the l-k-plane of Rn by angle α. It follows that the application of all possible matrices Sn(l,k;α) to x allows an orthogonal transformation in all n(n-1)/2 two-dimensional planes of Rn. For this reason, the product of all n(n-1)/2  n-dimensional Givens-matrices is defined by (73)Snα1,…,α1/2nn-1≔∏i=1n-1∏j=i+1nSni,j;α1/2i-12n-i+j-i.By now, it is possible to transform a random vector X~Φg,P,ϑ,Ψ orthogonally with given angles α1,…,αn(n-1)/2. The transformed contour defining polyhedron then represents a rotated but congruent representative of given P. The proof of the following theorem can be done analogously to that of Theorem 14.Theorem 16.
LetX~Φg,P,ϑ,Ψ. Given the angles α1,…,αn(n-1)/2, the random vector Z=Sn(α1,…,αn(n-1)/2)X is Φg,POT,μ,Σ distributed, where POT=Sn(α1,…,αn(n-1)/2)P, Σ=Sn(α1,…,αn(n-1)/2)ΨSn(α1,…,αn(n-1)/2)T, μ=Sn(α1,…,αn(n-1)/2)ϑ, and φg,POT(z)=C(g,P)g(hP(Sn(α1,…,αn(n-1)/2)Tz)) is the density of Z.For an illustration of how Theorem16 applies, see Example A.4 in the Appendix. In case n=2, the Givens rotation reduces to a case considered in [43] for a geometric parametrization of the two-dimensional Gaussian law.


## 8. Statistics in Location-Scale Families

### 8.1. Estimating and Testing

Estimating the parametersμ and Σ from a Φg,P,μ,Σ-distributed sample can be done using different methods. In a reasonable way, classical empirical moments (74)μ^=1k∑r=1kXr,Σ^=1k∑r=1kXr-μ^Xr-μ^Tturn out to be method of moments estimators. To find maximum-likelihood estimators of the lower triangular matrix V∈Rn×n with vii≠0, ∀i, and the vector ρ∈Rn in a Φg,PLT,Vθ+ρ,VΨVT-model, one has to numerically solve a nonlinear minimizing problem. Note that there are various algorithms introduced in the literature to solve such problems. To this end, we refer to [44–46] and for a global overview to [47], respectively. A concrete application of this method will be given below.Similarly, deriving maximum-likelihood estimates ofSn(α1,…,αn/2(n-1)) and ρ in a Φg,POT,Snα1,…,αn/2n-1μ,Snα1,…,αn/2n-1ΨSnα1,…,αn/2n-1 model needs to solve a slightly modified nonlinear problem. Moreover, robust parameter estimation following [48, 49] can be seen to successfully work in the present distribution class. An analogous remark holds true for the application in the following section.Now we discuss how to use one-dimensional Kolmogorov-Smirnov tests to verify if a dataset is distributed according to a given polyhedral star-shaped distribution. To this end, we use the polyhedral star-shaped polar coordinate transformation and Theorem5 from Section 3.1 to convert the random vector X~Φg,P,ϑ,Ψ into a tuple of random variables and applying the well-known Kolmogorov-Smirnov test to each random variable. Assuming a given realization (x1,…,xk) of independent and identically distributed random vectors X1,…,Xk,Xi∈Rn,i=1,…,k, we can apply the transformation SPOLP-1 for every i=1,…,k to the realization xi to generate polyhedral star-shaped polar coordinates ri,φi,1,…,φi,n-1,i=1,…,k. For a given polyhedral star-shaped distribution Φg,P,ϑ,Ψ and X~Φg,P,ϑ,Ψ we can apply Theorem 5 and test the empirical distribution functions of the realizations against the distributions of the random variables R,Φ1,…,Φn-1. This can be done by the Kolmogorov-Smirnov test statistics (75)Tkr=suptFkt,r-FRt,Tkφj=supαFkα,φj-FΦjα,j=1,…,n-1,where Fk(t,r) and Fk(α,φj) are the usual empirical distribution functions. Since we consider the marginal distributions we can reject our realization to be Φg,P,ϑ,Ψ distributed if the tests are rejected for one polar coordinate. Note that the integral representations of FR and FΦj, j=1,…,n-1 are difficult to compute, especially in high dimensions. To this end, we can use Algorithm 1 to simulate k~ independently, Φg,P,ϑ,Ψ distributed realizations (x~1,…,x~k~), apply Theorem 5 to (x~1,…,x~k~), replace the exact cdf FR and FΦj, j=1,…,n-1 by the empirical cdf Fk~(t,r~) and Fk~(α,φ~j), j=1,…,n-1, and perform two sample Kolmogorov-Smirnov tests with test statistics (76)Tk,k~r,r~=suptFkt,r-Fk~t,r~,Tk,k~φj,φ~j=supαFkα,φj-Fk~α,φ~j,j=1,…,n-1.


### 8.2. Application

To conclude our considerations we model a polyhedral star-shaped distribution based on real data using the methods we described in this paper. To be more specific, we consider premature mortality and the median household income in all counties of the state Georgia (USA) and model their joint distribution to describe how premature mortality is influenced by household income in that particular state. Doing this is inspired by the descriptive statistical analysis in [1].Premature mortality is defined as the all-cause, age-adjusted mortality rate for all individuals based on all deaths occurring before the age of 75 that could have been prevented. The data for the sample of all 159 counties in Georgia are from the CDC Compressed Mortality database (https://wonder.cdc.gov) and from the County Health Ranking of the USA (http://www.countyhealthrankings.org) and averaged form 2008 through 2011. For an illustration of the sample, see the scatter plot in Figure 9. Testing this sample for normality via Shapiro-Wilk test is significantly high rejected with a resulting p value p=1.516·10-7. Since the density of the bivariate normal distribution is characterized by elliptical level sets (see [43]), the intention to model the data is here to keep the multinormal dgf g(r)=e-r2/2 but to change the contours of the level sets using a specific star-shaped polygon. Inspired by the optical appearance of the data point cloud, we choose P to be the triangle defined by its vertices p1=(-7,-6)T, p2=(-11,19)T, and p3=(21,-6)T; see Figure 9. This results in the star-shaped polyhedral distribution Φg,P,ϑ,Ψ, where (77)ϑ=1.879974.3866,Ψ=203.13237-98.91335-98.91335141.42441,φg,P,ϑ,Ψx,y=1700e-1/2max⁡-25x/199-4y/199,25x/333+32y/333,-y/62,x,y∈R.For the modelling we divide the data randomly into two groups, one for the estimation and one for testing the estimated models. Applying the estimation methods from Section 8.1 for the first group of our data yields estimates that can be found in Table 2.Table 2
Model estimates.
Estimates Method of moments μ^=39.2799244.28225,Σ^=118.70887-61.49893-61.4989366.09405  Maximum-likelihood (a)ρ^=37.6556742.24069,V^=0.69571350-0.12851030.6720302 (b)ρ^=36.689642.0099,S2(α^)=cos⁡(0.1343)sin⁡(0.1343)-sin⁡(0.1343)cos⁡(0.1343)  M-estimation ρ ^ = 35.7269 42.7719 ,   S 2 ( α ^ ) = cos ⁡ ( 0.1437 ) sin ⁡ ( 0.1437 ) - sin ⁡ ( 0.1437 ) cos ⁡ ( 0.1437 )Figure 9
Median household income and premature mortality in the counties of Georgia, averaged from 2008 through 2011 (a) and contour defining triangleP (b).
(a) (b)Applying Theorem14 with the method of moments estimates and the maximum-likelihood estimates (a) yields Φg,PLT,μ^,Σ^ and Φg,PLT,V^ϑ+ρ^,V^ΨV^T, respectively. The application of Theorem 16 with the maximum-likelihood estimates (b) and the m-estimates, respectively, yields Φg,POT+ρ^,S2(α^)ϑ+ρ^,S2(α^)ΨS2(α^)T. We tested the second group of the data against these estimated models by using the two sample Kolmogorov-Smirnov tests from Section 8.1 with 105 simulated random points of the estimated models. The results are shown in Table 3. It turns out that the estimated models we generated using orthogonal transformation cannot be accepted as appropriate models of the data in this case. The choice of the contour defining polygon P was not good enough to model the data properly by using orthogonal transformations of Φg,P,ϑ,Ψ. Since the greatest p values of our tests are received with the maximum-likelihood estimates (a), we prefer Φg,PLT,V^ϑ+ρ^,V^ΨV^T as distribution model of the data, where (78)V^ϑ+ρ^=38.9635944.94702,V^ΨV^T=98.31958-64.40732-64.4073284.31032,φg,PLTx,y=0.003·exp12max⁡-0.1818x-0.0312y+8.161,-0.0449x-0.2585y+12.611,0.1313x+0.149y-11.23972,x,y∈R.For an illustration of the empirical cdf of the marginal distributions used for the two sample Kolmogorov-Smirnov tests of our preferred model, see Figure 10. The density and its level sets of the model are visualized in Figure 11. Since the value of the correlation of our estimated model is -0.7074 we can conclude a strong negative correlation between premature mortality and household income which is a first hint to that lower income results can be accompanied by higher premature mortality rates in Georgia. Furthermore, in a simulation of 106 independent Φg,PLT,V^ϑ+ρ^,V^ΨV^T distributed random points, we observed that 36.7% of the points were related to an income higher than 38.96359 and a mortality rate lower than 44.94702, and 36.2% of the points were related to an income lower than 38.96359 and a mortality rate higher than 44.94702. Even when choosing a tolerance level of a half standard deviation σ/2, the rates are approximately equal, namely, 20.3% (income higher than 38.96359+σ/2 and a mortality rate lower than 44.94702-σ/2) and 19.6% (income lower than 38.96359-σ/2 and a mortality rate higher than 44.94702+σ/2). This is a second hint to that a higher household income is accompanied by a lower premature mortality rate in Georgia.Table 3
Kolmogorov-Smirnov tests of the data against the estimated distribution models.
p value of the K. S. test for R p value of the K. S. test for Φ Method of moments 0.7782463 0.4186046  Maximum-likelihood (a)0.9205786 (a)0.6398323 (b)6.992589⋅10-8 (b)0.1791245  M-estimation 3.928482 ⋅ 1 0 - 8 0.2240325Figure 10
Simulated empirical cdf of the marginal distributions ofR (a) and Φ (b) of the second group of data and Φg,PLT,V^ϑ+ρ^,V^ΨV^T.
(a) (b)Figure 11
Density levels sets (a) and density (b) ofΦg,PLT,V^ϑ+ρ^,V^ΨV^T.
(a) (b)


## 8.1. Estimating and Testing

Estimating the parametersμ and Σ from a Φg,P,μ,Σ-distributed sample can be done using different methods. In a reasonable way, classical empirical moments (74)μ^=1k∑r=1kXr,Σ^=1k∑r=1kXr-μ^Xr-μ^Tturn out to be method of moments estimators. To find maximum-likelihood estimators of the lower triangular matrix V∈Rn×n with vii≠0, ∀i, and the vector ρ∈Rn in a Φg,PLT,Vθ+ρ,VΨVT-model, one has to numerically solve a nonlinear minimizing problem. Note that there are various algorithms introduced in the literature to solve such problems. To this end, we refer to [44–46] and for a global overview to [47], respectively. A concrete application of this method will be given below.Similarly, deriving maximum-likelihood estimates ofSn(α1,…,αn/2(n-1)) and ρ in a Φg,POT,Snα1,…,αn/2n-1μ,Snα1,…,αn/2n-1ΨSnα1,…,αn/2n-1 model needs to solve a slightly modified nonlinear problem. Moreover, robust parameter estimation following [48, 49] can be seen to successfully work in the present distribution class. An analogous remark holds true for the application in the following section.Now we discuss how to use one-dimensional Kolmogorov-Smirnov tests to verify if a dataset is distributed according to a given polyhedral star-shaped distribution. To this end, we use the polyhedral star-shaped polar coordinate transformation and Theorem5 from Section 3.1 to convert the random vector X~Φg,P,ϑ,Ψ into a tuple of random variables and applying the well-known Kolmogorov-Smirnov test to each random variable. Assuming a given realization (x1,…,xk) of independent and identically distributed random vectors X1,…,Xk,Xi∈Rn,i=1,…,k, we can apply the transformation SPOLP-1 for every i=1,…,k to the realization xi to generate polyhedral star-shaped polar coordinates ri,φi,1,…,φi,n-1,i=1,…,k. For a given polyhedral star-shaped distribution Φg,P,ϑ,Ψ and X~Φg,P,ϑ,Ψ we can apply Theorem 5 and test the empirical distribution functions of the realizations against the distributions of the random variables R,Φ1,…,Φn-1. This can be done by the Kolmogorov-Smirnov test statistics (75)Tkr=suptFkt,r-FRt,Tkφj=supαFkα,φj-FΦjα,j=1,…,n-1,where Fk(t,r) and Fk(α,φj) are the usual empirical distribution functions. Since we consider the marginal distributions we can reject our realization to be Φg,P,ϑ,Ψ distributed if the tests are rejected for one polar coordinate. Note that the integral representations of FR and FΦj, j=1,…,n-1 are difficult to compute, especially in high dimensions. To this end, we can use Algorithm 1 to simulate k~ independently, Φg,P,ϑ,Ψ distributed realizations (x~1,…,x~k~), apply Theorem 5 to (x~1,…,x~k~), replace the exact cdf FR and FΦj, j=1,…,n-1 by the empirical cdf Fk~(t,r~) and Fk~(α,φ~j), j=1,…,n-1, and perform two sample Kolmogorov-Smirnov tests with test statistics (76)Tk,k~r,r~=suptFkt,r-Fk~t,r~,Tk,k~φj,φ~j=supαFkα,φj-Fk~α,φ~j,j=1,…,n-1.


## 8.2. Application

To conclude our considerations we model a polyhedral star-shaped distribution based on real data using the methods we described in this paper. To be more specific, we consider premature mortality and the median household income in all counties of the state Georgia (USA) and model their joint distribution to describe how premature mortality is influenced by household income in that particular state. Doing this is inspired by the descriptive statistical analysis in [1].Premature mortality is defined as the all-cause, age-adjusted mortality rate for all individuals based on all deaths occurring before the age of 75 that could have been prevented. The data for the sample of all 159 counties in Georgia are from the CDC Compressed Mortality database (https://wonder.cdc.gov) and from the County Health Ranking of the USA (http://www.countyhealthrankings.org) and averaged form 2008 through 2011. For an illustration of the sample, see the scatter plot in Figure 9. Testing this sample for normality via Shapiro-Wilk test is significantly high rejected with a resulting p value p=1.516·10-7. Since the density of the bivariate normal distribution is characterized by elliptical level sets (see [43]), the intention to model the data is here to keep the multinormal dgf g(r)=e-r2/2 but to change the contours of the level sets using a specific star-shaped polygon. Inspired by the optical appearance of the data point cloud, we choose P to be the triangle defined by its vertices p1=(-7,-6)T, p2=(-11,19)T, and p3=(21,-6)T; see Figure 9. This results in the star-shaped polyhedral distribution Φg,P,ϑ,Ψ, where (77)ϑ=1.879974.3866,Ψ=203.13237-98.91335-98.91335141.42441,φg,P,ϑ,Ψx,y=1700e-1/2max⁡-25x/199-4y/199,25x/333+32y/333,-y/62,x,y∈R.For the modelling we divide the data randomly into two groups, one for the estimation and one for testing the estimated models. Applying the estimation methods from Section 8.1 for the first group of our data yields estimates that can be found in Table 2.Table 2
Model estimates.
Estimates Method of moments μ^=39.2799244.28225,Σ^=118.70887-61.49893-61.4989366.09405  Maximum-likelihood (a)ρ^=37.6556742.24069,V^=0.69571350-0.12851030.6720302 (b)ρ^=36.689642.0099,S2(α^)=cos⁡(0.1343)sin⁡(0.1343)-sin⁡(0.1343)cos⁡(0.1343)  M-estimation ρ ^ = 35.7269 42.7719 ,   S 2 ( α ^ ) = cos ⁡ ( 0.1437 ) sin ⁡ ( 0.1437 ) - sin ⁡ ( 0.1437 ) cos ⁡ ( 0.1437 )Figure 9
Median household income and premature mortality in the counties of Georgia, averaged from 2008 through 2011 (a) and contour defining triangleP (b).
(a) (b)Applying Theorem14 with the method of moments estimates and the maximum-likelihood estimates (a) yields Φg,PLT,μ^,Σ^ and Φg,PLT,V^ϑ+ρ^,V^ΨV^T, respectively. The application of Theorem 16 with the maximum-likelihood estimates (b) and the m-estimates, respectively, yields Φg,POT+ρ^,S2(α^)ϑ+ρ^,S2(α^)ΨS2(α^)T. We tested the second group of the data against these estimated models by using the two sample Kolmogorov-Smirnov tests from Section 8.1 with 105 simulated random points of the estimated models. The results are shown in Table 3. It turns out that the estimated models we generated using orthogonal transformation cannot be accepted as appropriate models of the data in this case. The choice of the contour defining polygon P was not good enough to model the data properly by using orthogonal transformations of Φg,P,ϑ,Ψ. Since the greatest p values of our tests are received with the maximum-likelihood estimates (a), we prefer Φg,PLT,V^ϑ+ρ^,V^ΨV^T as distribution model of the data, where (78)V^ϑ+ρ^=38.9635944.94702,V^ΨV^T=98.31958-64.40732-64.4073284.31032,φg,PLTx,y=0.003·exp12max⁡-0.1818x-0.0312y+8.161,-0.0449x-0.2585y+12.611,0.1313x+0.149y-11.23972,x,y∈R.For an illustration of the empirical cdf of the marginal distributions used for the two sample Kolmogorov-Smirnov tests of our preferred model, see Figure 10. The density and its level sets of the model are visualized in Figure 11. Since the value of the correlation of our estimated model is -0.7074 we can conclude a strong negative correlation between premature mortality and household income which is a first hint to that lower income results can be accompanied by higher premature mortality rates in Georgia. Furthermore, in a simulation of 106 independent Φg,PLT,V^ϑ+ρ^,V^ΨV^T distributed random points, we observed that 36.7% of the points were related to an income higher than 38.96359 and a mortality rate lower than 44.94702, and 36.2% of the points were related to an income lower than 38.96359 and a mortality rate higher than 44.94702. Even when choosing a tolerance level of a half standard deviation σ/2, the rates are approximately equal, namely, 20.3% (income higher than 38.96359+σ/2 and a mortality rate lower than 44.94702-σ/2) and 19.6% (income lower than 38.96359-σ/2 and a mortality rate higher than 44.94702+σ/2). This is a second hint to that a higher household income is accompanied by a lower premature mortality rate in Georgia.Table 3
Kolmogorov-Smirnov tests of the data against the estimated distribution models.
p value of the K. S. test for R p value of the K. S. test for Φ Method of moments 0.7782463 0.4186046  Maximum-likelihood (a)0.9205786 (a)0.6398323 (b)6.992589⋅10-8 (b)0.1791245  M-estimation 3.928482 ⋅ 1 0 - 8 0.2240325Figure 10
Simulated empirical cdf of the marginal distributions ofR (a) and Φ (b) of the second group of data and Φg,PLT,V^ϑ+ρ^,V^ΨV^T.
(a) (b)Figure 11
Density levels sets (a) and density (b) ofΦg,PLT,V^ϑ+ρ^,V^ΨV^T.
(a) (b)


## 9. Discussion and Outlook

In Sections8.1 and 8.2 we considered m-estimates assuming fully specified dgf g. Referring to Table 1 where density generating functions are given with unknown parameters, one could ask for a simultaneous estimation of these parameters and m-estimates. The simultaneous m-estimation of α1,…,αn(n-1)/2, ρ, and parameters of g failed so far by numerical problems. Solving this is postponed to future work.In Section8.1 we presented univariate Kolmogorov-Smirnov tests to test data against a given polyhedral star-shaped distribution Φg,P,ϑ,Ψ. Since with this method we test with univariate marginal distributions of Φg,P,ϑ,Ψ, it is an open problem to develop a multivariate test for the class of polyhedral star-shaped distributions to validate also the dependencies in the model. Doing this, however, is postponed to future work.



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*Source: 7176897-2017-02-14.xml*