A unified method for generating closed-form point estimators for exponential families: An example with the beta distribution applied to proportions of land used for farming

We show that, after a simple power-transform reparameterization of the (vector) exponential family, the solutions to the likelihood equations coincide with moment-type estimating equations. This equivalence enables a unified route to closed-form point estimators for multi-parameter models that typically lack explicit maximum likelihood (ML) solutions. Within this framework we (i) recover, as special cases, several recent closed-form estimators from the literature; (ii) derive new families of estimators indexed by monotone transformations $g$; and (iii) establish strong consistency and asymptotic normality under mild regularity, including a dominated differentiation condition. As a detailed illustration, we derive closed-form estimators for parameters that index the beta distribution. A Monte Carlo simulation study is carried out to evaluate and compare the performance of proposed and existing estimators. Finally, we illustrate the approach with a novel municipal data set regarding proportions of land used for farming in Roraima (Brazil), which, to the best of our knowledge, has not been analyzed in the literature before, demonstrating the method's practical usefulness.