Polyhedral Star-Shaped Distributions

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.

Simulation of polyhedral convex contoured distributions

“…Our starting point is the consideration of the uniform distribution on a convex polyhedron in the latter sections. In Section 5.1 we summarize certain stochastic and geometric representations and linear transformation methods from (Richter 2014;2015a) and (Richter and Schicker 2017) for the particular classes of convex polyhedra and polyhedral convex contoured distributions, respectively. Specific applications of representations (6) and (4) below are presented in Sections 5.2 and 5.3, respectively.…”

“…where G(A) = {ϑ ∈ R n−1 : ∃η = η(ϑ)with (ϑ T , η) T ∈ A} and (B n ∩ S) + denotes the Borel σ -field on the upper half-sphere of S. For the proof of (5) and further integral representations of the star-generalized surface measure of star-shaped polyhedra, we refer to (Richter and Schicker 2017). Let us recall that since 0 n ∈ int(P) it is possible to calculate the Minkowski functional h P of P as described in Section 2.2.…”

“…For further specific geometric representation formulae of polyhedral star-shaped distributions, having star-shaped polyhedra as contour defining polyhedron, see again (Richter and Schicker 2017). For the choice of a dgf g there are various possibilities.…”

“…For an overview about this broad field of research, we refer to (Arnold et al 2008;Balkema et al 2010;Fang et al 1990;Fernandez and Osiewalski 1995;Gupta and Song 1997;Kamiya et al 2008;Richter 2009;2013;2014;2015a;2015b;Richter and Schicker 2017;Sarabia and Gomez-Deniz 2008). Convex polyhedral distributions are characterized by contours being the topological boundaries of convex polyhedra and can be considered being a subclass of polyhedral star-shaped distributions that are studied in (Richter and Schicker 2017). To simulate them needs just the additional independent simulation of a positive random variable playing the role of a certain generalized radius variable of a generalized ball.…”

“…Comparisons of the general acceptance-rejection algorithm with particular algorithms from the literature and applications to drug combination studies are given in Section 4. A general technique from (Richter 2014(Richter , 2015aRichter and Schicker 2017) is summarized in Section 5.1 to construct the general class of polyhedral convex contoured distributions from the particular polyhedral convex contoured uniform distribution. Sections 5.2 and 5.3 deal with applications of the presented geometric and stochastic representations to simulating probabilities of random events and to linear transformations, respectively.…”