A transformed-score approach to closed-form and one-step efficient estimation for the beta distribution

Abstract

Power transformations of beta random variables produce unbiased estimating equations from transformed-model likelihood scores. This construction clarifies the connection between moment-type and likelihood-based estimating equations, recovers recent closed-form estimators for the beta shape parameters, and yields a new family indexed by a scalar transformation parameter $r$. For each fixed admissible $r$, we establish strong consistency and joint asymptotic normality. We also show that likelihood-guided selection over a prespecified finite grid preserves root-$n$ consistency and use one Fisher-scoring update to obtain an asymptotically efficient estimator. A Monte Carlo study with 1000 replications compares numerical maximum likelihood, two existing closed-form procedures, the proposed likelihood-selected closed-form estimator, and its one-step refinement across nine parameter configurations and four sample sizes. All procedures are evaluated on the same simulated samples within each replication. The selected closed-form estimator closely tracks maximum likelihood, while the one-step refinement is nearly indistinguishable from it for moderate sample sizes. Finally, the methods are illustrated descriptively using the proportions of municipal area devoted to crops and pasture in the 15 municipalities of Roraima, Brazil.

40 pages, 3 figures