The uniform distribution on the simplex
The continuous uniform distribution is a basic probability distribution defined over an interval on the real line. Its density function is a constant on that interval and 0 elsewhere. The multivariate continuous uniform distribution often appears in the literature defined over an n-dimensional rectangle, hence the name of rectangular distribution. However, it can be defined over more general bounded regions with finite measure. One of the most common applications of this model is the random numbers generation, used in uniform experimental designs.In this work we propose to adapt the continuous multivariate uniform distribution on the simplex over different bounded regions as compositional rectangles, compositional hexagons, Borel or convex sets. We will provide its density function with respect to the Aitchison measure, we will study its characteristics measures and its main algebraic properties with special emphasis on those related to the algebraic geometric structure of the simplex.
Keywords: multiuniform distribution convex set Aitchison measure