  This study presents new closed-form estimators for the Dirichlet and the
Multivariate Gamma distribution families, whose maximum likelihood estimator
cannot be explicitly derived. The methodology builds upon the score-adjusted
estimators for the Beta and Gamma distributions, extending their applicability
to the Dirichlet and Multivariate Gamma distributions. Expressions for the
asymptotic variance-covariance matrices are provided, demonstrating the
superior performance of score-adjusted estimators over the traditional moment
ones. Leveraging well-established connections between Dirichlet and
Multivariate Gamma distributions, a novel class of estimators for the latter is
introduced, referred to as "Dirichlet-based moment-type estimators". The
general asymptotic variance-covariance matrix form for this estimator class is
derived. To facilitate the application of these innovative estimators, an R
package called estimators is developed and made publicly available.
