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

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-24-2026-282
Published: Sep 2, 2026
Cite as:
Dalia Kamal Alnagar, Classical and Bayesian Estimation of Weibull Distribution Parameters with an Application to Daily Wind Speed Data, Int. J. Anal. Appl., 24 (2026), 282.

Main Article Content

Dalia Kamal Alnagar

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.

Article Details

References

  1. C.G. Justus, W.R. Hargraves, A. Mikhail, D. Graber, Methods for Estimating Wind Speed Frequency Distributions, J. Appl. Meteorol. 17 (1978), 350–353. https://doi.org/10.1175/1520-0450(1978)017<0350:mfewsf>2.0.co;2.
  2. J.V. Seguro, T.W. Lambert, Modern Estimation of the Parameters of the Weibull Wind Speed Distribution for Wind Energy Analysis, J. Wind Eng. Ind. Aerodyn. 85 (2000), 75–84. https://doi.org/10.1016/s0167-6105(99)00122-1.
  3. S.A. Akdağ, A. Dinler, A New Method to Estimate Weibull Parameters for Wind Energy Applications, Energy Convers. Manag. 50 (2009), 1761–1766. https://doi.org/10.1016/j.enconman.2009.03.020.
  4. I. Usta, An Innovative Estimation Method Regarding Weibull Parameters for Wind Energy Applications, Energy 106 (2016), 301–314. https://doi.org/10.1016/j.energy.2016.03.068.
  5. A. Gelman, J.B. Carlin, H.S. Stern, D.B. Dunson, A. Vehtari, D.B. Rubin, Bayesian Data Analysis, 3rd ed., Chapman and Hall/CRC, 2013. https://doi.org/10.1201/b16018.
  6. A.N. Celik, A Statistical Analysis of Wind Power Density Based on the Weibull and Rayleigh Models at the Southern Region of Turkey, Renew. Energy 29 (2004), 593–604. https://doi.org/10.1016/j.renene.2003.07.002.
  7. H. Teimourian, M. Abubakar, M. Yildiz, A. Teimourian, A Comparative Study on Wind Energy Assessment Distribution Models: A Case Study on Weibull Distribution, Energies 15 (2022), 5684. https://doi.org/10.3390/en15155684.
  8. A. Bulut, O. Bingöl, Weibull Parameter Estimation Methods on Wind Energy Applications - a Review of Recent Developments, Theor. Appl. Climatol. 155 (2024), 9157–9184. https://doi.org/10.1007/s00704-024-05184-2.
  9. S. Rehman, T.O. Halawani, T. Husain, Weibull Parameters for Wind Speed Distribution in Saudi Arabia, Sol. Energy 53 (1994), 473–479. https://doi.org/10.1016/0038-092x(94)90126-m.
  10. A.H. Alanazi, F.A. Almutlaq, Assessment of Wind Energy Potential at Three Prime Locations in Saudi Arabia: Analysis of Sharma, Qurayyat and Sakaka Sites, Am. J. Nat. Sci. 5 (2024), 30–48. https://doi.org/10.47672/ajns.2074.
  11. K.S. AlQdah, R. Alahmdi, A. Alansari, A. Almoghamisi, M. Abualkhair, M. Awais, Potential of Wind Energy in Medina, Saudi Arabia Based on Weibull Distribution Parameters, Wind Eng. 45 (2021), 1652–1661. https://doi.org/10.1177/0309524x211027356.
  12. M.I. Faraz, Strategic Analysis of Wind Energy Potential and Optimal Turbine Selection in Al-Jouf, Saudi Arabia, Heliyon 10 (2024), e39188. https://doi.org/10.1016/j.heliyon.2024.e39188.
  13. E. Köksal Babacan, S. Kaya, Comparison of Parameter Estimation Methods in Weibull Distribution, Sigma J. Eng. Nat. Sci. 38 (2020), 1609–1621.
  14. J. Wang, J. Hu, K. Ma, Wind Speed Probability Distribution Estimation and Wind Energy Assessment, Renew. Sustain. Energy Rev. 60 (2016), 881–899. https://doi.org/10.1016/j.rser.2016.01.057.
  15. V. Sohoni, S. Gupta, R. Nema, A Comparative Analysis of Wind Speed Probability Distributions for Wind Power Assessment of Four Sites, Turk. J. Electr. Eng. Comput. Sci. 24 (2016), 4724–4735. https://doi.org/10.3906/elk-1412-207.
  16. H.E. Akyuz, H. Gamgam, Statistical Analysis of Wind Speed Data with Weibull, Lognormal and Gamma Distributions, Cumhuriyet Sci. J. 38 (2017), 68–76. https://doi.org/10.17776/csj.358773.
  17. T. Soukissian, Use of Multi-Parameter Distributions for Offshore Wind Speed Modeling: The Johnson S_B Distribution, Appl. Energy 111 (2013), 982–1000. https://doi.org/10.1016/j.apenergy.2013.06.050.
  18. P. Lencastre, A. Yazidi, P.G. Lind, Modeling Wind-Speed Statistics beyond the Weibull Distribution, Energies 17 (2024), 2621. https://doi.org/10.3390/en17112621.
  19. M. Kamal, M. Alam, O.A. Alqasem, E.I. Mohamed, M.I. Habadi, D.K. Alnagar, R. Aldallal, A.S. Alghamdi, Estimation and Statistical Properties of the Weibull–Rayleigh Model with Applications to Industrial Data, Alexandria Eng. J. 128 (2025), 556–570. https://doi.org/10.1016/j.aej.2025.05.001.