A study on comparisons of Bayesian and classical parameter estimation methods for the two-parameter Weibull distribution

Yıl 2020, Cilt: 69 Sayı: 1, 576 - 602, 30.06.2020

Asuman Yılmaz Mahmut Kara Halil Aydoğdu

https://doi.org/10.31801/cfsuasmas.606890

Cited By: 2

Öz

The main objective of this paper is to determine the best estimators of the shape and scale parameters of the two parameter Weibull distribution. Therefore, both classical and Bayesian approximation methods are considered. For parameter estimation of classical approximation methods maximum likelihood estimators (MLEs), modified maximum likelihood estimators-I (MMLEs-I), modified maximum likelihood estimators -II (MMLEs-II), least square estimators (LSEs), weighted least square estimators (WLSEs), percentile estimators (PEs), moment estimators (MEs), L-moment estimators (LMEs) and TL- moment estimators (TLMEs) are used. Since the Bayesian estimators don't have the explicit form. There are Bayes estimators are obtained by using Lindley's and Tierney Kadane's approximation methods in this study. In Bayesian approximation, the choice of loss function and prior distribution is very important. Hence, Bayes estimators are given based on both the non- informative and informative prior distribution. Moreover, these estimators have been calculated under different symmetric and asymmetric loss functions. The performance of classical and Bayesian estimators are compared with respect to their biases and MSEs through a simulation study. Finally, a real data set taken from Turkish State Meteorological Service is analysed for better understanding of methods presented in this paper.

Anahtar Kelimeler

Bayes approximation, , parameter estimation, , new estimator, L- moment estimator, simulation study

Kaynakça

• Lawless, J. F., Statistical Models and Methods for Life time Data, 3rd Edition, John Wiley and Sons, New York 2003.
• Tang, Y., Extended Weibull Distributions in Reliability Engineering, A Thesis Submitted to Department of Industrial System Engineering, National University of Singapore, 2004.
• Justus, C. G., et al., Methods for estimating wind speed frequency distributions, Journal of applied meteorology, 17(3), (1978) 350-353.
• Kantar, Y. M., Şenoglu, B., (2008). A comparative study for the location and scale parameters of the Weibull distribution with given shape parameter, Computers Geosciences, 34(12), (2008), 1900-1909.
• Genc, A., et al. Estimation of wind power potential using Weibull distribution, Energy Sources, 27(9), (2005), 809-822.
• Stevens, M. J. M., and Smulders, P. T., The estimation of the parameters of the Weibull wind speed distribution for wind energy utilization purposes, Wind engineering, (1979), 132-145.
• Araki, Junichi, et al., Weibull distribution function for cardiac contraction: integrative analysis, American Journal of Physiology-Heart and Circulatory Physiology, 277(5), (1999), 1940-1945.
• Fernandez, A., et al., Application of a frequency distribution model to describe the thermal inactivation of two strains of Bacillus cereus, Trends in food science technology, 10(4), (1999), 158-162.
• Zobeck, Ted M., Thomas, E. Gill, and Thomas, W. Popham, A two-parameter Weibull function to describe airborne dust particle size distributions, Earth Surface Processes and Landforms: The Journal of the British Geomorphological Research Group, 24(10), (1999), 943-955.
• Trustrum, K., Jayatilaka, A. S., On estimating the Weibull modulus for a brittle material, Journal of Material Science, 14, (1979),1080-1084.
• Hung, Wen-Liang, Weighted least-squares estimation of the shape parameter of the Weibull distribution, Quality and Reliability Engineering International, 17(6 ) (2001), 467-469.
• Lu, Hai-Lin, Chen, Chong-Hong and Wu, Jong-Wuu, A note on weighted least-squares estimation of the shape parameter of the Weibull distribution. Quality and Reliability Engineering International, 20.6 (2004), 579-586.
• Pobocâkova, Ivana, and Sedliackova, Zuzana, Comparison of four methods for estimating the Weibull distribution parameters, Appl. Math. Sci. 8(1), (2014), 4137-4149. Teimouri, Mahdi, Hoseini, Seyed M. and Nadarajah, Saralees, Comparison of estimation methods for the Weibull distribution. Statistics 47(1) (2013), 93-109.
• Alizadeh, M., Rezaei, S. and Bagheri, S. F., On the estimation for the Weibull distribution, Annals of Data Science, 2(4) (2015), 373-390.
• Al Omari, M.A. and Ibrahim, N.A., Bayesian survival estimator for Weibull distribution with censored Data, J. Appl. Sci. 11, (2011), 393-396.
• Guure, Chris Bambey, Ibrahim, Noor Akma and Al Omari, Mohammed Ahmed, Bayesian estimation of two-parameter weibull distribution using extension of Jeffreys' prior information with three loss functions, Mathematical Problems in Engineering 2012 (2012).
• Pandey, B.N., Dwividi, N. and Pulastya, B., Comparison between Bayesian and maximum likelihood estimation of the scale parameter in Weibull distribution with known shape under linex loss function, J. Sci. Res., 55, (2011), 163-172.
• Al-Athari, F.M., Parameter estimation for the double-Pareto distribution, J. Mathematics Statistics, 7 (2011), 289-294.
• Zellner, Arnold, Bayesian estimation and prediction using asymmetric loss functions, Journal of the American Statistical Association, 81(394), (1986), 446-451.
• Kundu, D. and Raqab, M.Z., Generalized Rayleigh distribution: Different methods of estimations, Comput. Statist.Data Anal., 49 (2005), 187-200.
• Ramos, Pedro Luiz, et al., The Frechet distribution: Estimation and Application an Overview, arXiv preprint arXiv 2018:1801.05327.
• Dey, Sanku, et al., Two-parameter Maxwell distribution: Properties and different methods of estimation, Journal of Statistical Theory and Practice, 10(2) (2016), 291-310.
• Alkasasbeh, M.R. and Raqab, M.Z., Estimation of the generalized logistic distribution parameters: Comparative study, Statist. Methodol. 6, (2009), 262-279.
• Howard, R., Antle, C. and Klimko, L.A., Maximum likelihood estimation with the Weibull model, Journal of the American Statistical Association, 69(345) (1974), 246-249.
• Cohen, A.C., Maximum likelihood estimation in the Weibull distribution based on complete and on censored samples, Technometrics, 7(4) (1965), 579-588.
• Swain, James J., Venkatraman, Sekhar and Wilson, James R., Least-squares estimation of distribution functions in Johnson's translation system, Journal of Statistical Computation and Simulation 29(4) (1988), 271-297.
• Kao, John H.K., Computer methods for estimating Weibull parameters in reliability studies, IRE Transactions on Reliability and Quality Control, (1958), 15-22.
• Kao, John H.K., A graphical estimation of mixed Weibull parameters in life-testing of electron tubes, Technometrics, 1(4) (1959), 389-407.
• Hosking, Jonathan R.M, L-moments: Analysis and estimation of distributions using linear combinations of order statistics, Journal of the Royal Statistical Society: Series B (Methodological) 52(1) (1990), 105-124.
• Abdul-Moniem, I. B., TL-moments and L-moments estimation for the Weibull distribution, Advances and Application in Statistics, 15(1 ) (2009), 83-99.
• Elamir, Elsayed A.H, and Seheult, Allan H, Trimmed L-moments, Computational Statistics Data Analysis, 43(3 ) (2003), 299-314.
• Cohen, C.A., and Whitten, B., Modified maximum likelihood and modified moment estimators for the three-parameter Weibull distribution. Communications in Statistics-Theory and Methods, 11(23) (1982), 2631-2656.
• Box Cep, Tiag, G. C., Bayesian inference in statistical analysis, 1973.
• Renjini, K. R., Abdul-Sathar, E. I. and Rajesh, G. A study of the effect of loss functions on the Bayes estimates of dynamic cumulative residual entropy for Pareto distribution under upper record values, Journal of Statistical Computation and Simulation, 86(2) (2016), 324-339.
• Ali, Sajid, Aslam, Muhammad and Kazmi, Syed Mohsin Ali, A study of the effect of the loss function on Bayes Estimate, posterior risk and hazard function for Lindley distribution, Applied Mathematical Modelling, 37(8), (2013), 6068-6078.
• Legendre, A., New Methods for the Determination of Orbits of Comets Courcier, Paris, 1805.
• Gauss, C. F., Least squares method for the combinations of observations (translated by J. Bertrand 1955), Mallet-Bachelier, Paris, 1810.
• Feynman, Richard P., Mr. Feynman goes to Washington, Engineering and Science, 51(1), (1987), 6-22.
• Calabria, R. and Pulcini, G., An engineering approach to Bayes estimation for the Weibull distribution, Micro Electron Reliab, 34, (1994), 789-802.
• Norstrom, Jan Gerhard, The use of precautionary loss functions in risk analysis, IEEE Transactions on reliability, 45(3), (1996), 400-403.
• Helu, Amal and Samawi, Hani, The Inverse Weibull Distribution as a Failure Model Under Various Loss Functions and Based on Progressive First-Failure Censored Data, Quality Technology Quantitative Management, 12(4), (2015), 517-535.
• Ali, Sajid, On the mean residual life function and stress and strength analysis under different loss function for Lindley distribution, Journal of Quality and Reliability Engineering, 2013.
• O'Hagan, A., and Luce, B. R., A primer on Bayesian statistics in health economics and outcomes research, Sheffield: Centre for Bayesian Statistics in Health Economics, 2003.
• Acquah, H. D., Bayesian Logistic Regression Modelling via Markov Chain Monte Carlo Algorithm, Journal of Social and Development Sciences, 4(4), (2013), 193-197.
• Lindley, Dennis V., Approximate Bayesian methods. Trabajos de estadstica y de investigacin operativa, 31(1), (1980), 223-245.
• Tierney, Luke, and Kadane, Joseph B., Accurate approximations for posterior moments and marginal densities, Journal of the American statistical association, 81(393), (1986), 82-86.
• Singh, S. K., Singh, U. and Yadav, A. S. Bayesian estimation of Marshall-Olkin extended exponential parameters under various approximation techniques, Hacettepe Journal of Mathematics and Statistics, 43(2), (2014), 347-360.