Abstract
A new method of probabilistic modelling of polyhedrally contoured sample clouds is presented and applied to statistical reasoning for a real dataset. Various representations of the new class of polyhedral star-shaped distributions are derived and basic properties of the moments as well as characteristic and moment generating functions of these distributions are studied. Along with location-scale transformations, estimating and hypothesis testing are dealt with.
1. Introduction
One of nowadays challenges of statistical modelling is the construction of flexible multivariate probability distributions given a dataset. In [1], disparities in premature mortality between high- and low-income US counties are dealt with on the basis of descriptive statistics. For a subsequent step of statistical reasoning, a method of constructing a suitable probabilistic model is needed. Analyzing the correlation between mortality and income cannot be done in a common linear regression model because of absence of homoscedasticity. The present paper provides therefore a new method of constructing flexible multivariate distribution laws being well adapted to practical problems characterized by polyhedral contours of their sample clouds. Moreover, we will particularly suggest a specific model and further statistical reasoning for the premature mortality-income data mentioned above.
Several basic methods and results from the research area of constructing multivariate distributions are surveyed, for example, in [2] and the numerous papers mentioned there. Multivariate densities with given contours were already introduced in [3] and further studied in [4, 5]. Constructing star-shaped distributions on using Minkowski functionals and generalized uniformly distributed random vectors on generalized spheres, proving geometric measure representations of such distributions and stochastic representations of correspondingly distributed random vectors are to be found in [6]. Constructions and representations in the special cases of norm and antinorm contoured distributions are recently dealt with in [7]. Numerous applications of geometric measure representations are surveyed in the last mentioned two papers and in [8]. In [9] more recent applications to extreme values are presented. More general order statistics are dealt with elsewhere.
The aim of the present paper is to deal with another important special case of star-shaped distributions where the contours are the topological boundaries of polyhedra.
Polyhedral star-shaped distributions can be considered being a subclass of the class of star-shaped distributions. Before studying specific properties of polyhedral star-shaped distributions, we start therefore with a short introduction to the general theory of star-shaped distributions. To this end, we follow [6].
Let denote a star body, that is, a nonempty and compact star-shaped set being equal to the closure of its interior and having the origin in its interior. The functional defined byis known as the Minkowski functional of . Here, we assume that is positive homogeneous; that is, , , and consider and its boundary being the star ball and star sphere of Minkowski radius , respectively. Since unambiguously defines the considered star ball, it is possible to study subclasses of star bodies by specifying . One can choose, for example, to be a norm or an antinorm. For the latter notion we refer to [10]. Here, specific representations of are considered later in this paper, if denotes a star-shaped polyhedron. For simplicity, consideration will be restricted throughout this paper to star bodies having the following property.
A countable collection of pairwise disjoint cones with vertex being the origin and is called a fan. Let , , where denotes the Borel--field in , and . In what follows, the star body and a set are chosen such that for every the set is well defined and for every there is uniquely determined satisfying .
A function satisfying the assumptions where is called a density generating function (dgf),a star-shaped density and its contour defining star body. Such densities are studied in [3, 6, 11]. A probability measure having the density will be denoted by . Note that the normalizing constant allows the representationwhere denotes the Lebesgue measure in . For examples of density generating functions, see [12] or Table 1. Moreover, the definition of specific star-shaped densities appears already in earlier work where is prespecified. If is the Euclidean-norm, one considers the class of spherically symmetric distributions; see [12–14] and many other contributions. The class of convex contoured -symmetric distributions is considered if is the -norm, . This class of distributions was introduced in [15] and studied, for example, in [16–19]. In [20] stochastic and geometric representations are derived also for the case which can be called according to [10] the antinorm contoured case. General norm contoured distributions in are studied in [8] by choosing in terms of a norm in . Minkowski functionals of ellipsoids and even -generalized ellipsoids are used in the (re-)construction of common and -generalized elliptically contoured distributions in [21] and in [6], respectively.
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 [6–8, 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 [28–30].
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.
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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 [32–34]. 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.
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Theorem 11. If then its characteristic function allows the representation
Proof. Applying the stochastic representation (52) and Theorem in [38], it is proved in Theorem in [6] thatApplying the representation of in Remark (a) of [6] and the polyhedral star-shaped polar coordinate transformation to (64) yields (63).
Note that if is symmetrically distributed with respect to the origin, , the imaginary part in equation (63) vanishes. To be specific, let follow the -star contoured distribution . The characteristic function of is 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 .
5.3. Moment Generating Functions
Considering a random vector the moment generating function can be formally represented byIt is possible to specify representation (66) given a certain type of dgf and a certain representation of . The results are shown in the following theorem.
Theorem 12. Let and the special Kotz-type dgf , be given. If is represented by Lemma 2 (b1) and it exists an such that for all is the Minkowski functional of the star-shaped polyhedron then
Proof. Since it holds
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 Theorem 8 and (52) offer the possibility to generate polyhedral star-shaped distributed random points for a given contour defining star-shaped polyhedron and a dgf . By generating a random number from the density and independently of this a polyhedral star-shaped generalized uniformly distributed random point on , the product gives the desired polyhedral star-shaped distributed random point. can be simulated by generating a uniformly on distributed random point and applying Theorem 8; that is, . It remains the simulation of . To this end, we use a cuboid satisfying 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 -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.
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The sampling method in step (1) has to be specified in accordance with the chosen dgf . For the dgfs in Table 1, sampling methods can be found in [12]. Proving that is uniformly distributed on and the stopping time of step (2) is finite can be done analogously to Appendix of [29]. A suitable cuboid can be found as follows. If is represented by , one can putwhere denotes the th component of the vertex . If is represented by , one can solve for every , the optimization problems: minimize (maximize ) subject to . Since it appears to be easier to find a suitable in case of , note that it is possible to transform from to according to [40, 41].
Example 13. Let be -star contoured distributed, with the Kotz-type dgf where . Figures 4, 5, and 6 show 2000 points simulated independently according to .
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7. Location-Scale Transformations of Continuous Polyhedral Star-Shaped Distributions
Throughout this section, we assume that has a finite second-order moment and use the notation , where and denote expectation vector and covariance matrix of , respectively. We consider location and scale transformations of and study transformed density level sets.
Theorem 14. Let with symmetric and positive definite matrix . If is another symmetric and positive definite matrix and then there exist regular lower triangular matrices and from such that , and the random vector follows the distribution , where , and the density is represented as
Proof. For the existence of and , see [42]. Furthermore, , and . The representation of follows by the density transformation formula.
For an example of the application of the linear transformation in Theorem 14, 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 into a polyhedron being generally not congruent with . Referring to this, see again Figures 7 and 8. Since noncongruent change of shape is not always a desired property, because 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].
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Definition 15. Let and . The -dimensional -Givens matrix is defined by Note that, for every dimension , there exist Givens matrices. A Givens rotation of a vector defined by is an orthogonal transformation and rotates within the --plane of by angle . It follows that the application of all possible matrices to allows an orthogonal transformation in all two-dimensional planes of . For this reason, the product of all -dimensional Givens-matrices is defined by By now, it is possible to transform a random vector orthogonally with given angles . The transformed contour defining polyhedron then represents a rotated but congruent representative of given . The proof of the following theorem can be done analogously to that of Theorem 14.
Theorem 16. Let . Given the angles , the random vector is distributed, where , , , and is the density of .
For an illustration of how Theorem 16 applies, see Example A.4 in the Appendix. In case , 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 parameters and from a -distributed sample can be done using different methods. In a reasonable way, classical empirical moments turn out to be method of moments estimators. To find maximum-likelihood estimators of the lower triangular matrix with , , and the vector in a -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 of and in a 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 Theorem 5 from Section 3.1 to convert the random vector into a tuple of random variables and applying the well-known Kolmogorov-Smirnov test to each random variable. Assuming a given realization of independent and identically distributed random vectors , we can apply the transformation for every to the realization to generate polyhedral star-shaped polar coordinates . For a given polyhedral star-shaped distribution and we can apply Theorem 5 and test the empirical distribution functions of the realizations against the distributions of the random variables . This can be done by the Kolmogorov-Smirnov test statistics where and are the usual empirical distribution functions. Since we consider the marginal distributions we can reject our realization to be distributed if the tests are rejected for one polar coordinate. Note that the integral representations of and , are difficult to compute, especially in high dimensions. To this end, we can use Algorithm 1 to simulate independently, distributed realizations , apply Theorem 5 to , replace the exact cdf and , by the empirical cdf and , , and perform two sample Kolmogorov-Smirnov tests with test statistics
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 value . 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 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 to be the triangle defined by its vertices , , and ; see Figure 9. This results in the star-shaped polyhedral distribution , where 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.
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Applying Theorem 14 with the method of moments estimates and the maximum-likelihood estimates (a) yields and , respectively. The application of Theorem 16 with the maximum-likelihood estimates (b) and the m-estimates, respectively, yields . 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 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 was not good enough to model the data properly by using orthogonal transformations of . Since the greatest values of our tests are received with the maximum-likelihood estimates (a), we prefer as distribution model of the data, where 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 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 independent distributed random points, we observed that of the points were related to an income higher than and a mortality rate lower than , and of the points were related to an income lower than and a mortality rate higher than . Even when choosing a tolerance level of a half standard deviation , the rates are approximately equal, namely, (income higher than and a mortality rate lower than ) and (income lower than and a mortality rate higher than ). This is a second hint to that a higher household income is accompanied by a lower premature mortality rate in Georgia.
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9. Discussion and Outlook
In Sections 8.1 and 8.2 we considered m-estimates assuming fully specified dgf . 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 , , and parameters of failed so far by numerical problems. Solving this is postponed to future work.
In Section 8.1 we presented univariate Kolmogorov-Smirnov tests to test data against a given polyhedral star-shaped distribution . Since with this method we test with univariate marginal distributions of , 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.
Appendix
Example A.1. Let be a pentagon given by the vertices , , , , and and a convex polyhedral fan for , where , , and ; see Figure 12. Applying Lemma 2 (b1), where , , , , and the Minkowski functional of can be represented as Let and .
Note that can be calculated applying (i)the polyhedral star-shaped polar coordinate representation as (ii)the star-spherical coordinate representation as (iii)the facet-content representation by calculating , , , , , , and (iv)the simplicial representation as Applying the definition of the polyhedral star-shaped generalized uniform distribution (42) we can calculate by . Note that it is also possible to calculate this ratio using the sector measure and cone measure, respectively, since .
Example A.2. In this example we consider again the class of -stars from Section 4.2. Apart from the representation of in the mentioned section it is also possible to apply Lemma 2 (b2) and represent by where and are for all the elements of the matrix and , where and , are equivalent to those in Section 4.2. Since it follows Thus we can calculate the first and second vector of moments by
Example A.3. Let be the tetragon defined by the vertices , , , and . Considering the convex polyhedral fan for , where as shown in Figure 7(a), we can represent applying Lemma 2 (b1) as , .
Assuming the dgf , it follows and , since . The random vector follows the density and first- and second-order moments are If we wish to transform by a linear transformation into a random vector where denotes the two-dimensional identity matrix, we can apply Theorem 14. It follows then , where can be calculated by Cholesky decomposition of ; thus then follows the density For an illustration of the transformed contour defining tetragon and the density level sets of the transformed density, see Figure 8.
Example A.4. Assume that the polyhedron , the dgf , and the random vector are the same as in Example A.3. Applying Theorem 16 with it follows that is distributed, and the resulting density function allows the representation For an illustration of the transformed contour defining tetragon and the density level sets of the transformed density, see Figure 13.
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Competing Interests
The authors declare further that there is no conflict of interests regarding the publication of this paper.