Abstract
The class of (p1,…,pk)-spherical probability laws and a method of simulating random vectors following such distributions are introduced using a new stochastic vector representation. A dynamic geometric disintegration method and a corresponding geometric measure representation are used for generalizing the classical χ2-, t- and F-distributions. Comparing the principles of specialization and marginalization gives rise to an alternative method of dependence modeling.
Highlights
1 Introduction A basic notion from the theory of spherical probability laws is that of the stochastic basis which is a random vector following the uniform distribution on the Euclidean unit sphere in the k-dimensional Euclidean space Rk, see e.g. Fang et al (1990)
Multivariate uniform distributions are introduced in an algebraic way without referring to any type of surface measure
Numerous authors deal with the uniform distribution by considering the density of its k − 1-dimensional marginal distribution, an approach which will, not further be discussed, here
Summary
A basic notion from the theory of spherical probability laws is that of the stochastic basis which is a random vector following the uniform distribution on the Euclidean unit sphere in the k-dimensional Euclidean space Rk, see e.g. Fang et al (1990). The point of view of uniformity of which is the speech here is to define it by having a constant Radon-Nikodym derivative with respect to the Euclidean surface content measure This geometric view onto the class of spherical distributions is the background of the corresponding geometric measure representation (2) in Richter (1991). The coordinates mentioned were introduced in Richter (2007) solving a long standing problem apparently conclusively treated as insolvable in Szablowski (1998) It is a natural step of research to consider random vectors having p-spherical uniform distributions with positive components of p =
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