Regression for Compositions based on a Generalization of the Dirichlet Distribution

Consider a positive random vector following a compound distribution where the compounding parameter multiplies non-random scale parameters. The associated composition is the vector divided by the sum of its components. The conditions under which the composition depends on the distribution of the compounding parameter are given. When the original vector follows a compound distribution based on independent Generalized Gamma components, the Simplicial Generalized Beta (SGB) is the most general distribution of the composition that is invariant with respect to the distribution of the compounding parameter. 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. Distributional checks are made possible through the marginal distributions of functions of the parts that should be Beta distributed. A multiple SGB regression procedure is set up and applied to data from the United Kingdom Time Use survey.

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