A General Class of Distributions on the Simplex

Abstract

SUMMARY The Dirichlet class of distributions on the simplex is overstructured for practical work because of its many strong independence properties. A persistent problem of distribution theory has thus been to extend this class to include members free of these independence properties. An alternative approach has been the recent introduction of the logistic-normal class which contains members with and without the independence properties. Unfortunately, this class is separate from the Dirichlet class and so cannot sustain discussion of some strong forms of independence. These difficulties are overcome in this paper by the introduction of a more general class which includes as special cases the Dirichlet and logistic-normal classes. The role of this new class in relation to the analysis of compositional data, in particular the investigation of independence hypotheses, is discussed and illustrated.