A distribution on the simplex of the Generalized Beta type

Consider a random vector with positive components following a compound distribution where the compounding parameter multiplies fixed scale parameters. The closed random vector 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 unrelatedness of the distribution of the closed random vector to the compounding distribution depends on the parameters of the generalized Gamma. This fact is exemplified with the multivariate Generalized Beta distribution of the second kind (MGB2) in which the compounding parameter follows an inverse Gamma distribution. We call the most general distribution of the closed random vector, for which the compounding 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.