Classical and Bayesian Estimation of Weibull Distribution Parameters with an Application to Daily Wind Speed Data

Abstract

Accurate characterization of the probability distribution of wind speed is essential for preliminary wind resource assessment and renewable energy planning. This paper presents a comparative classical and Bayesian analysis of daily wind speed distributions using 24 years (2001–2024, \(n = 8{,}766\) daily observations) of NASA POWER daily mean wind speed data at 10 m above the surface for Tabuk, Saudi Arabia. The two-parameter Weibull distribution, a standard benchmark model in wind-energy studies, is fitted using three classical estimators—maximum likelihood estimation (MLE), the method of moments (MoM), and L-moments—alongside a Bayesian analysis with weakly informative Gamma priors on the shape and scale parameters, in which posterior inference is carried out via a random-walk Metropolis–Hastings Markov chain Monte Carlo (MCMC) algorithm and convergence is assessed using the Gelman–Rubin diagnostic across multiple chains. The MLE and Bayesian posterior-mean estimates of the Weibull parameters agree closely (\(\hat{k} = 3.553\), \(\hat{\lambda} = 4.235\) m/s), reflecting the dominance of the likelihood in this large sample. A Monte Carlo simulation study further shows that the Bayesian estimator attains a lower mean squared error than the MLE for the shape parameter in small samples (\(n \leq 100\)). Goodness-of-fit comparisons against Rayleigh, Gamma, and Lognormal alternatives show that the Lognormal and Gamma distributions provide a better formal statistical fit to the Tabuk daily wind speed data than the Weibull distribution, confirming that the Weibull model should not be assumed statistically optimal for every wind regime, even though it remains a useful and interpretable benchmark for wind-energy applications. Using the fitted Weibull model, the mean wind power density at 10 m height is estimated at approximately 43.9 W/m\(^2\), indicating a low near-surface wind resource. The results provide a statistically grounded characterization of the Tabuk wind regime, illustrate the practical trade-offs between classical and Bayesian Weibull estimation, and underscore the importance of benchmarking the Weibull distribution against alternative probability models before drawing conclusions about wind-resource potential.