A new distribution on the simplex containing the Dirichlet family

The Dirichlet family owes its privileged status within simplex distributions to easynessof interpretation and good mathematical properties. In particular, we recall fundamentalproperties for the analysis of compositional data such as closure under amalgamationand subcomposition. From a probabilistic point of view, it is characterised (uniquely)by a variety of independence relationships which makes it indisputably the referencemodel for expressing the non trivial idea of substantial independence for compositions.Indeed, its well known inadequacy as a general model for compositional data stemsfrom such an independence structure together with the poorness of its parametrisation.In this paper a new class of distributions (called Flexible Dirichlet) capable of handlingvarious dependence structures and containing the Dirichlet as a special case is presented.The new model exhibits a considerably richer parametrisation which, for example,allows to model the means and (part of) the variance-covariance matrix separately.Moreover, such a model preserves some good mathematical properties of the Dirichlet,i.e. closure under amalgamation and subcomposition with new parameters simplyrelated to the parent composition parameters. Furthermore, the joint and conditionaldistributions of subcompositions and relative totals can be expressed as simple mixturesof two Flexible Dirichlet distributions.The basis generating the Flexible Dirichlet, though keeping compositional invariance,shows a dependence structure which allows various forms of partitional dependenceto be contemplated by the model (e.g. non-neutrality, subcompositional dependenceand subcompositional non-invariance), independence cases being identified by suitableparameter configurations. In particular, within this model substantial independenceamong subsets of components of the composition naturally occurs when the subsetshave a Dirichlet distributionGeologische Vereinigung; Institut d’Estadística de Catalunya; International Association for Mathematical Geology; Càtedra Lluís Santaló d’Aplicacions de la Matemàtica; Generalitat de Catalunya, Departament d’Innovació, Universitats i Recerca; Ministerio de Educación y Ciencia; Ingenio 2010