  There exist several endeavors proposing a new family of extended
distributions using the beta-generating technique. This is a well-known
mechanism in developing flexible distributions, by embedding the cumulative
distribution function (cdf) of a baseline distribution within the beta
distribution that acts as a generator. Univariate beta-generated distributions
offer many fruitful and tractable properties and have applications in
hydrology, biology and environmental sciences amongst other fields. In the
univariate cases, this extension works well, however, for multivariate cases,
the beta distribution generator delivers complex expressions. In this document,
the proposed extension from the univariate to the multivariate domain addresses
the need of flexible multivariate distributions that can model a wide range of
multivariate data. This new family of multivariate distributions, whose
marginals are beta-generated distributed, is constructed with the function
H(x_{1},...,x_{p})=F(G_{1}(x_{1}),G_{2}(x_{2}),...,G_{p}(x_{p})), where
$G_{i}(x_{i})$ are the cdfs of the gamma (baseline) distribution and F(.) as
the cdf of the Dirichlet distribution. Hence as the main example, a general
model having the support [0,1]^{p} (for p variates), using the Dirichlet as the
generator, is developed together with some distributional properties, such as
the moment generating function. The proposed Dirichlet-generated distributions
can be applied to compositional data. The parameters of the model are estimated
by using the maximum likelihood method. The effectiveness and prominence of the
proposed family are illustrated by analyzing simulated as well as two real
datasets. A new model testing technique is introduced to evaluate the
performance of the multivariate models.
