Frequentist Versus Bayesian Parameter Estimation of The Beta Distribution: A Simulation-Based Analysis
Year: 2026
Type: article
Abstract: The Beta distribution is a highly useful continuous probability distribution defined on the interval (0, 1) and characterized by two shape parameters, α and β. Accurate estimation of these parameters is crucial in various fields, ranging from reliability engineering to Bayesian A/B testing and click-through rate smoothing in data science. This paper presents a comprehensive comparative analysis of three parameter estimation frameworks: the Method of Moments (MOM), Maximum Likelihood Estimation (MLE), and Bayesian estimation via Maximum A posteriori (MAP) under a Gamma prior. Building on previous investigations which demonstrated the superiority of MOM in small sample sizes (N<50), this study conducts extensive Monte Carlo simulations across multiple scenarios (symmetric, skewed, and U-shaped distributions) and sample sizes ranging from 10 to 500. Our results confirm that MOM remains computationally trivial and has lower mean squared error (MSE) than MLE in extremely small samples (N=10), while MLE is asymptotically efficient and excels in large samples. The Bayesian MAP estimator with a Gamma(2, 0.5) prior outperforms MLE in small-to-moderate samples in the skewed and symmetric scenarios, with MSE reductions of roughly 90% in the most favourable case, and converges to MLE as the sample size increases. A sensitivity analysis over six Gamma priors shows, however, that this gain depends on how well the prior is located relative to the true parameters: poorly located priors give little improvement or are worse than MLE, most clearly in the U-shaped scenario. Convergence diagnostics over 21,000 simulated datasets show no optimizer failures or boundary solutions for MLE or MAP and no variance violations for MOM. We discuss these trade-offs and provide concrete recommendations for practitioners.
DOI: 10.14419/qf3g7k08
Institutions: American University of Nigeria, University of Nigeria
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7225084316