A distribution on the simplex of the Generalized Beta type
Résumé |
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. |
Mots-clés |
Dirichlet distribution; Generalized Beta distribution of the second kind; simplicial Generalized Beta; maximum likelihood estimation; imputation. |
Citation | Graf, M. (2018). A distribution on the simplex of the Generalized Beta type. Presented at Seminar, Department of Statistics at Universidad Carlos III of Madrid. |
Type | Présentation (Anglais) |
Date | 18-5-2018 |
Evénement | Seminar (Department of Statistics at Universidad Carlos III of Madrid) |
Liée au projet | Convention Université de Neuchâtel/Office fédéral de la s... |