High-dimensional star-shaped distributions

Wolf-Dieter Richter

doi:10.1186/s40488-019-0096-0

Wolf-Dieter Richter

Open Access

https://doi.org/10.1186/s40488-019-0096-0

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Abstract

Stochastic representations of star-shaped distributed random vectors having heavy or light tail density generating function g are studied for increasing dimensions along with corresponding geometric measure representations. Intervals are considered where star radius variables take values with high probability, and the derivation of values of distribution functions of g-robust statistics is proved to be based upon considering random events whose probability is asymptotically negligible if the dimension of the sample vector is approaching infinity. Moreover, a principal component representation of p-generalized elliptically contoured p-generalized Gaussian distributions is discussed.

Highlights

Among the frequently obtained impressions one gets from analyzing high-dimensional data sets are that an observation point’s distance from the zero element of the sample space is likely to belong to a certain interval from the positive real line, away from zero, and that the distribution of the direction of the vector seems to be close, in a certain sense, to a uniform distribution on the set of all directions that are observable from a certain center
In situations of the described type, it may be reasonable to model the data, or their residuals after fitting to a model, by multivariate star-shaped distributions. In this regard, (Balkema and Embrechts 2007) and (Balkema et al 2010) discover conditions ensuring that star-shaped distributions with the Gauss-exponential law being one of the most known examples appear as limit laws in certain high-risk scenarios
Distributions from the class of star-shaped distributions are flexible with respect to convexity or radial concavity, allow different variability of probability mass along different directions of the sample space and are able to model light and heavy distribution centers and tails

Summary

Introduction

Among the frequently obtained impressions one gets from analyzing high-dimensional data sets are that an observation point’s distance from the zero element of the sample space is likely to belong to a certain interval from the positive real line, away from zero, and that the distribution of the direction of the vector seems to be close, in a certain sense, to a uniform distribution on the set of all directions that are observable from a certain center. “A measure concentration property” section is aimed to consider typical intervals where R takes values if X is star-shaped distributed, and in “On g-robust statistics” section distributions of univariate statistics are described which can basically be derived from star-uniformly distributed vectors. Such distributions are not affected by whether X has a density generating function g generating light or heavy distribution tails and is called g-robust. The example deals with the asymptotic behavior of star surface content and volume of star spheres and star balls or ellipsoids, respectively, if dimension is approaching infinity.

Note that

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The proof is finished by observing that

The statistic