Journal of Probability and Statistics

Volume 2015 (2015), Article ID 165468, 12 pages

http://dx.doi.org/10.1155/2015/165468

## Convex and Radially Concave Contoured Distributions

Institute of Mathematics, University of Rostock, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany

Received 24 June 2015; Revised 18 October 2015; Accepted 21 October 2015

Academic Editor: Z. D. Bai

Copyright © 2015 Wolf-Dieter Richter. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Integral representations of the locally defined star-generalized surface content measures on star spheres are derived for boundary spheres of balls being convex or radially concave with respect to a fan in . As a result, the general geometric measure representation of star-shaped probability distributions and the general stochastic representation of the corresponding random vectors allow additional specific interpretations in the two mentioned cases. Applications to estimating and testing hypotheses on scaling parameters are presented, and two-dimensional sample clouds are simulated.

#### 1. Introduction

The families of multivariate Gaussian and elliptically contoured distributions have served for a long time as the main basis of numerous probabilistic models and their many successful applications. Basics of estimation theory of elliptically contoured distributions can be found in [1–3] and, for example, [4]. Advancing needs of statistical practice as well as longstanding challenging mathematical questions stimulated the development of larger classes of probability laws containing many well known distributions as particular elements. We note that [5] surveys a big part of the distribution theory in . Numerous authors contributed to establishing the class of multivariate star-shaped distributions. For a recent review of this development, see [6]. Estimating level sets of star-shaped densities has been dealt with in [7–10].

Several aspects of analyzing a cloud of sample points may be of importance for the process of defining a class of probability laws. The visual impression of the appearance of star-shaped figures built by the points of a sample cloud may lead to the idea that the boundaries of star-bodies, henceforth called star-spheres, represent density level sets of a probability law. Counting the sample points belonging to thin layers about star-spheres then leads to the idea that a certain function that assigns a nonnegative number to every such star-sphere serves as the density generating function (dgf) of a nonnegative random variable (rv) or, more generally, as a function being proportional to the Radon-Nikodym density of a multivariate probability law with respect to a certain -finite measure defined on the sample space. We call such a function a univariate level density function of the multivariate probability distribution.

The combination of the aspects of defining level sets of a multivariate density and of assigning a nonnegative level to every such set will be reflected here in a new method of integration. This method may be considered as a heightening and generalization of the classical principle of Cavalieri which was modified by Torricelli. Combining integration on level sets with that along the levels may also be considered as a geometric disintegration method. This method is essentially based upon certain non-Euclidean surface measures on star-spheres. It is one of the main aims of this paper to further develop this theory for two important types of star-spheres. Convex bodies and star-bodies being radially concave with respect to a fan in build these two classes of star-bodies. Thus, in this paper, the focus is on considering probability laws having the boundary of such sets as their density level sets or their contour sets. Actually, the results in Section 3 are mainly restricted to, possibly shifted, symmetric contour sets being norm or antinorm spheres.

There are different ways to introduce a dfg of a continuous probability law. Looking through the statistical and mathematical literature, one finds many interesting nonnegative and suitably integrable functions which may serve as a dfg. Another way to introduce a dfg is to analyze the structure of a known multivariate density and to extract from it, if possible, a function which does not depend on the surface measure on the star-spheres but depends exclusively on the levels of the multivariate density.

It is well known that the definition of a dfg is not unique and so is how to deal with this circumstance. Densities with heavy tails may be of interest in (re)insurance, and densities with light tails may be of interest in reliability theory. Both types of densities can be modeled using each time a suitable dfg.

Modeling the density level sets and the univariate level density of a multivariate distribution can be done in a combined way or separately from each other. Sometimes a parameter may influence both the level density and the density contour sets of a multivariate distribution. Another parameter may be only for one of these two aspects of importance.

The paper is organized as follows. Some basic facts from the theory of star-shaped distributions, with an emphasis on geometric measure representations, are collected in Section 2. New geometric descriptions of surface measures on boundary spheres of balls being radially concave with respect to a fan in , or convex, are presented in Section 3. Moreover, new statistical applications of geometric measure representations of norm and antinorm contoured distributions and of stochastic representations of correspondingly distributed random vectors are discussed there. In particular, distributions are illustrated by simulated sample clouds. The basics for estimating and testing hypothesis on scaling parameters are presented at the end of Section 3. Section 4 deals with proving the new results, and a discussion of the results can be found in the final Section 5.

#### 2. Star-Shaped Distributions

Geometric measure representations and stochastic representations of corresponding random vectors have been proved in [6] for general star-shaped distributions making essentially use of the notion of a star-generalized surface content measure. The latter is defined in a local way by taking derivatives of sector volumes and is known to be equivalently defined in an integral (in dimension two even explicitly differential geometric) way in the cases of -spheres and ellipsoids. For recent results and a survey of their probabilistic and statistical applications we refer to [6]. Here, some basic facts from star-shaped distribution theory and its applications will be summarized.

Let a random vector follow the probability density function (pdf): where is a vector of location, is a star-body having the origin as an inner point, is the distance function, or Minkowski functional, of the star-body ,the function satisfies , where , and the normalizing constant allows the representationAssuming that the technical Assumption 1 in [6] is satisfied which deals with a certain smoothness property of the boundary of , denotes the star-generalized surface content measure defined on the Borel subsets of . The probability measure corresponding to allows the geometric measure representation or disintegration formula: is called the density contour defining star-body of this distribution and any under consideration is a density generating function (dfg). The sets with may be considered playing the role of the indivisibles within a generalized principle of Cavalieri (which was modified by Torricelli). The random vector satisfies the stochastic representation:where and are stochastically independent, has the pdfand follows the star-generalized uniform probability distribution on the Borel--field over , , andBecause of (7), is called the star-generalized uniform basis of . The symbol means that the random vectors and follow the same probability law while indicates that the random vector follows the probability distribution .

For , we introduce the central projection cone,and the star sector of star radius ,whereis a star ball of star radius . Let be the Lebesgue measure in Then the star-generalized surface measure is defined on in a local approach by

Making use of the star-sphere intersection proportion function (ipf) of a set ,the disintegration representation of may be written as

The most immediate applications of this formula appear in cases where the ipf is a constant or an indicator of an interval. If, for a certain set , the ipf takes a constant value, , say, then is just equal to this value .

If, for a statistic , , , and the ipf of all sets take the constant value, , say, then the statistic is robust with respect to the dfg ; that is, the distribution of does not depend on .

If, for a certain set , the ipf is the indicator function of an interval, , say, thenFor specific statistical examples of such type we refer to Section 3. Applications of (13) in cases where the ipf is more structured are often more involved. Such a situation will be considered in Example 7.

The main aim of this paper, however, is not only to give attractive examples where the geometric measure representation applies but also to give nontrivial explanations of the locally defined surface measure on the basis of an integral (or differential geometric) approach. This will be done in the first two parts of Section 3 for the two important cases where is a norm or antinorm ball. As a result, in formulas (4), (5), (7) and (13), will afterwards allow additional specific integral (or differential geometric) interpretations in the two mentioned cases. The classwhere means the interior of , is called the class of continuous star-shaped distributions. A random vector is said in [6] to belong to the bigger class of star-shaped distributions if there are a vector , a star-body with (and boundary ), and a nonnegative random variable (rv) with cumulative distribution function (cdf) such that where and and are independent. In this case, we write The random vector is called the star-generalized uniform basis of the class . If is symmetric with respect to the origin, , then the functional is a norm, , if is convex, and is an antinorm, , if is radially concave with respect to a fan in For the latter notions, see [11]. Note that -symmetric distributions are norm or antinorm contoured if or , respectively. We will study general convex or norm contoured distributions in Section 3.1 and distributions being radially concave with respect to a fan in , or antinorm contoured, in Section 3.2. The main aim of these two sections is to give closer descriptions of being basic for both the general stochastic representation of in (5) and the specific geometric measure representations of in (7) and in (4) and (13) in case has a density. Moreover, two-dimensional distributions are illustrated by graphics showing simulated sample clouds. Applications to estimating and testing hypotheses on scaling parameters are demonstrated in Section 3.3. The proofs of the results from Sections 3.1 and 3.2 will be presented in Section 4, and a final discussion of the results follows in Section 5.

#### 3. Results

We start the presentation of new results with a remark on asymmetric distribution laws which seems to be very useful: a distribution being star-shaped with respect to a fan may be restricted to arbitrary unions of elements of .

*Remark 1. *Let , and . Thenis a probability law on .

Here, is a suitable index set. The proof of this result follows immediately by conditioning.

We call the collection of all such distributions the class of fan restricted star laws and denote it by :Elements of this distribution class are not symmetric, in general.

##### 3.1. Norm Contoured Distributions

Let be convex and symmetric with respect to the origin throughout this section. Our consideration is restricted therefore here to the class of norm contoured distributions:Let the system of Borel sets from the upper half of the sphere be . For , putand denote, wherever it exists, the outer normal vector to the norm sphere at the point by . Wherever the outer normal vector is not defined, let denote the zero element of Note that the set of boundary points of where does not exist is countable and thus without any influence on the value of the integral in the following theorem. We recall that the surface content measure was locally defined in (11).

Theorem 2. *In formulas (4), (5), (7) and (13), the surface content measure allows the representationwhere is the unit ball of the norm being dual to .*

We will refer to this result as to the integral or differential geometric approach to measuring surface content on a norm sphere based upon the dual norm geometry. We mention that a similar representation of follows for arbitrary . Due to Theorem 2, if is convex, the surface measure henceforth allows both the local and the integral interpretation in formulas (4), (5), (7) and (13). Moreover, Theorem 2 reflects a certain specific aspect of duality theory for norms.

In the next section we will deal in an analogous way with balls being radially concave with respect to a fan in

Figures 1–3 show sample clouds of size of -generalized Gaussian distributed two-dimensional () random vectors for different choices of , Notice that the six frames reflect different scaling of the clouds due to different values of . While the sample cloud in Figure 1(a) might seem to be similar to the illustration of the Gaussian case the shape of the sample cloud approaches that of an axes-aligned square if increases (or even tends to infinity). At the same time, the cloud (probability mass) becomes more and more concentrated. If, however, is tending to one then the shape of the sample cloud approaches that of the diamond. At the same time, probability mass becomes much less concentrated and the contour of the sample cloud appears to be not as sharp as in the opposite case. Hence, the parameter of such a distribution might be called a shape-concentration parameter. Note that Figures 6–8 also present sample clouds of convex contoured distributions but where emphasis is, inter alia, on the effect forced by an increasing sample size .