Dependence properties of generalized Liouville distributions on the simplex

Compositional data arise naturally in several branches of science, including chemistry, geology, biology, medicine, ecology, and manufacturing design. Thus the correct statistical analysis of this type of data is of fundamental importance. Prior to the pioneering and extensive work of Aitchison, the Dirichlet distribution provided the parametric model of choice when analyzing such data. But Aitchison and others have since pointed out that the Dirichlet distribution is appropriate only for modeling compositional vectors that exhibit forms of extreme independence. Aitchison developed his logistic normal classes partly in response to this shortcoming. Unfortunately, Aitchison’s logistic normal classes do not contain the Dirichlet distribution as a special case. As a result, they exhibit interesting dependence structures but are unable to model extreme independence. The generalized Liouville family is studied in this article. This family, which contains the Dirichlet class, is shown to contain densities that can model either complicated dependence or complicated independence structures.