Uniform distributions in a class of convex polyhedrons with applications to drug combination studies

Motivated by experimental designs for drug combination studies, in this paper, we propose a novel approach for generating a uniform distribution on an arbitrary tetragon in two-dimensional Euclidean space R². The key idea is to construct a one-to-one transformation between an arbitrary tetragon and the unit square [0, 1]². This transformation then provides a stochastic representation (SR) for the random vector uniformly distributed on the tetragon. An algorithm is proposed for generating a uniform distribution in an arbitrary triangular prism in R³. In addition, we develop methods for generating uniform distributions in a class of convex polyhedrons in n-dimensional Euclidean space Rⁿ. In particular, SRs for uniform distributions in regions with order restrictions are presented. We apply the proposed method to the experimental design for a drug combination study.