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.

翻译：本研究针对极大似然估计无法显式推导的Dirichlet分布族与多元Gamma分布族，提出了新的闭式估计量。该方法基于Beta分布与Gamma分布的得分调整估计量，将其适用性扩展到Dirichlet分布与多元Gamma分布。研究给出了渐近方差-协方差矩阵的表达式，证明了得分调整估计量相较于传统矩估计量的优越性能。利用Dirichlet分布与多元Gamma分布之间已得到充分验证的关联性，引入了一类新型的多元Gamma分布估计量，称为"基于Dirichlet的矩型估计量"，并推导了该类估计量的一般渐近方差-协方差矩阵形式。为促进这些创新估计量的实际应用，我们开发并公开了名为estimators的R语言包。