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

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 [

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

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

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

Geometric measure representations and stochastic representations of corresponding random vectors have been proved in [

Let a random vector

For

Making use of the star-sphere intersection proportion function (ipf) of a set

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

If, for a statistic

If, for a certain set

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

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

Let

Here,

We call the collection of all such distributions the class of fan restricted star laws and denote it by

Let

In formulas (

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

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

Figures

Sample clouds of convex contoured

Convex contoured cases far from the normal one.

Two extremal convex cases.

Finally, let us remark that Figure

Throughout the present section, let

for every

for every

In formulas (

In addition to the general local definition in formula (

Figures

This section deals with several examples where formula (

Let

We consider independent rv

While Examples

Let us assume that

Let

Applications of formula (

A nonconstant ipf of the set

For data in [

This example, which was presented by the authors only for the purpose of showing the use of the functions implemented by them to fit a linear regression model if errors are possibly exponentially power distributed, gives rise to throwing up in a similar two-dimensional situation the following standard question of statistical practice: how large should a sample size be to make a practical decision based upon a visual inspection “relatively safe”? In particular, how large should a sample size be for the observer being able to visually choose between the two two-dimensional

This question, clearly, is not formulated in a strong mathematical way and will not be answered in such way, here. Instead, we present Figures

Let

Moreover, let the random variable

If

We remark additionally that Figures

Finally, we notice that Figures

Sample clouds of radially concave contoured

Far reaching tails.

Generalized normal distribution: (small) sample size

Sample (medium) size

Sample (large) size

Two-layer

Independent realizations of two-layer

The general method of proof in this paper can be divided into two main parts. Using the properties of the support function of a convex body,

The support function of the convex body

Recall that if

According to Lemma 1 in [

We consider

The antisupport function of

The radial function of the radially concave star-shaped set

To make both the similarity and the difference between Theorems

In the applications of Theorem

The set

We show that if

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is grateful to Klaus Müller for his help in drawing the figures during the necessarily fast revision process.