A distribution on the simplex of the Generalized Beta type

Consider a random vector with positive components following a compound distribution where the mixing parameter multiplies fixed scale parameters. The closed random vector - or composition - is the vector divided by the sum of its components. We explicit on what conditions the distribution of the closed random vector does not depend on the mixing distribution. When the original vector has independent generalized Gamma components, it is shown that the invariance of the distribution of the closed random vector with respect to the mixing distribution depends on the parameters of the generalized Gamma components. This fact is exemplified with the multivariate Generalized Beta distribution of the second kind (MGB2) in which the mixing parameter follows an inverse Gamma distribution. We call the most general distribution of the closed random vector, for which the mixing parameter has no influence, the simplicial Generalized Beta (SGB). Some properties and moments of the SGB are derived. Conditional moments given a sub-composition give a way to impute missing parts when knowing a sub-composition only. Maximum likelihood estimators of the parameters are obtained. The method is applied to several examples.

Key words: Dirichlet distribution; Generalized Beta distribution of the second kind; simplicial Generalized Beta; maximum likelihood estimation; imputation.