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The simplicial generalized beta distribution. R-package and applications
A generalization of the Dirichlet and the scaled Dirichlet distributions is given by the simplicial generalized Beta, SGB (Graf, 2017). In the Dirichlet and the scaled Dirichlet distributions, the shape parameters are modeled with auxiliary variables (Maier, 2015, R-package DirichletReg) and Monti et al. (2011), respectively. On the other hand, in the ordinary logistic normal regression, it is the scale composition that is made dependent on auxiliary variables. The modeling of scales seems easier to interpret than the modeling of shapes. Thus in the SGB regression: - The scale compositions are modeled in the same way as for the logistic normal regression, i.e. each auxiliary variable generates D parameters, where D is the number of parts. - The D Dirichlet shape parameters, one for each part in the compositions, are estimated as well. - An additional overall shape parameter is introduced in the SGB that proves to have important properties in relation with non essential zeros. - Use of survey weights is an option. - Imputation of missing parts is possible. An application to the United Kingdom Time Use Survey (Gershuny and Sullivan, 2017) shows the power of the method. The R-package SGB (Graf, 2019) makes the method accessible to users. See the package vignette for more information and examples.
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.
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.
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.