Hypersphere candidates emanating from the Dirichlet and its extension

Abstract

Compositional datasets consist of observations that are proportional and are subject to non-negativity and unit-sum constraints. These datasets arise naturally in a multiplicity of fields such as agriculture, archaeology, economics, geology, health sciences, and psychology. There is a strong footprint in the literature on the Dirichlet distribution for modelling compositional datasets, followed by several generalizations of the Dirichlet distribution, with more flexible structures. In this study, we consider a transformation of two Dirichlet-type random variables W1,W2, . . . ,Wm by applying the square-root transformation Xi = √Wi for i = 1, 2, . . . , m. With this square-root transformation, we propose and develop a new distribution that is defined on the positive orthant of the hypersphere, that accommodates both positive and negative covariance structure. This novel model is a flexible offering to the spherical-Dirichlet models. We perform several simulation studies for the proposed model. The maximum likelihood is used for parameter estimation. Two applications of the models to biological and archaeological compositional datasets are presented, to illustrate the flexibility of the proposed model.