# An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method

• Talal Alharbi Department of Mathematic, College of Science, Qassim University Buraydah, Kingdom of Saudi Arabia
• Farag Hamad University of Benghazi, Libya
Keywords: Weibull distribution; maximum likelihood estimation; Newton-Raphson algorithm; root mean square error; Akaike information criterion; Bayesian information criterion.

### Abstract

Several methods have been used to estimate the Weibull parameters such as least square method (LSM), weighted least square method (WLSM), method of moments (MOM), and maximum likelihood (MLE). The maximum likelihood method is the most popular method (MLE). Newton-Raphson method has been applied to solve the normal equations of MLE’s in order to estimate the Weibull parameters. The method was used to find the optimal values of the Weibull distribution parameters for which the log-likelihood function is maximized. We tried to find the approximation solution to the normal equations of the MLE’s because there is no close form for get analytical solution. In this work, we tried to carry out a study that show the difference between two strategies to solve the MLE equations using Newton-Raphson algorithm. Both two strategies are provided an optimal solution to estimate the Weibull distribution parameters but which one more easer and which one converges faster. Therefore, we applied both strategies to estimate the Weibull’s shape and scale parameters using two different types of data (Real and simulation). We compared between the results that we got by applying the two strategies. Two studies have been done for comparing and selecting the optimal strategy to estimate Weibull distribution parameters using maximum likelihood method. We used some measurements to compare between the results such as number of steps for convergence (convergence condition), the estimated values for AIC, BIC and the RMSE value. The results show the numerical solution that we got by applying first strategy convergence faster than the solution that we got by applying second strategy. Moreover, the MRSE estimated by applying the first strategy is lower than the MRSE estimated by applying second strategy for the simulation study with different noise levels and different samples size.

### References

C. B. Guure and N. A. Ibrahim, “Methods for estimating the 2-parameter Weibull distribution with type-I censored data,” Res. J. Appl. Sci. Eng. Technol., vol. 5, no. 3, pp. 689–694, 2013.

D. Mao and W. Li, “A bounded derivative method for the maximum likelihood estimation on Weibull parameters,” ArXiv Prepr. ArXiv09064823, 2009.

P. Bhattacharya and R. Bhattacharjee, “A study on Weibull distribution for estimating the parameters,” J. Appl. Quant. Methods, vol. 5, no. 2, pp. 234–241, 2010.

C. Justus, W. Hargraves, A. Mikhail, and D. Graber, “Methods for estimating wind speed frequency distributions,” J. Appl. Meteorol., vol. 17, no. 3, pp. 350–353, 1978.

F. George, “A comparison of shape and scale estimators of the two-parameter Weibull distribution,” J. Mod. Appl. Stat. Methods, vol. 13, no. 1, p. 3, 2014.

H. Hirose, “Maximum likelihood estimation in the 3-parameter weibull distribution: a look through the generalized extreme-value distribution,” IEEE Trans. Dielectr. Electr. Insul., vol. 14, no. 1, pp. 257–260, 2007.

S. A. Czepiel, “Maximum likelihood estimation of logistic regression models: theory and implementation,” Available Czep Netstatmlelr Pdf, vol. 83, 2002.

T. A. Johansen, “Introduction to nonlinear model predictive control and moving horizon estimation,” Sel. Top. Constrained Nonlinear Control, vol. 1, pp. 1–53, 2011.

F. Hamad and N. N. Kachouie, “A hybrid method to estimate the full parametric hazard model,” Commun. Stat.-Theory Methods, vol. 48, no. 22, pp. 5477–5491, 2019.

S. Abbas, G. Ozal, S. H. Shahbaz, and M. Q. Shahbaz, “A new generalized weighted Weibull distribution,” Pak. J. Stat. Oper. Res., pp. 161–178, 2019.

L. A. Baharith, “New Generalized Weibull Inverse Gompertz Distribution: Properties and Applications,” Symmetry, vol. 16, no. 2, p. 197, 2024.

F. Hamad, N. Younus, M. Muftah, and M. Jaber, “Viability of transplanted organs based on Donor’s age,” Sch J Phys Math Stat, vol. 4, pp. 97–104, 2023.

F. Hamad and N. N. Kachouie, “Potential Impact of Donors’ Factors on Survival Times of Transplanted Hearts and Lungs,” Transplant. Rep., vol. 4, no. 4, p. 100035, 2019.

N. N. Kachouie and F. Hamad, “Potential impact of OPTN plan in association with donors’ factors on survival times of transplanted kidneys,” Transplant. Rep., vol. 5, no. 2, p. 100042, 2020.

D. Pinheiro, F. Hamad, M. Cadeiras, R. Menezes, and N. Nezamoddini-Kachouie, “A data science approach for quantifying spatio-temporal effects to graft failures in organ transplantation,” presented at the 2016 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), IEEE, 2016, pp. 3433–3436.

A. Kurniawan, N. Avicena, and E. Ana, “Estimation of the shape parameter of Weibull distribution based on type II censored data using EM algorithm,” AIP Publishing LLC, 2020, p. 030011.

N. Dibal, H. Bakari, and A. Yahaya, “Estimating the parameters in the two-parameter weibull model using simulation study and real-life data,” IOSR J. Math., vol. 12, pp. 38–42, 2016.

H. Hirose, “Maximum likelihood estimation in the 3-parameter weibull distribution: a look through the generalized extreme-value distribution,” IEEE Trans. Dielectr. Electr. Insul., vol. 14, no. 1, pp. 257–260, 2007.

F. Hamad, “Parametric Methods for Analysis of Survival Times with Applications to Organ Transplantation,” 2019.

N. Elmesmari, “Parameters Estimation Sensitivity of the Linear Mixed Model To Alternative Prior Distribution Specifications,” Sch J Phys Math Stat, vol. 9, pp. 166–170, 2021.

N. Balakrishnan and M. Kateri, “On the maximum likelihood estimation of parameters of Weibull distribution based on complete and censored data,” Stat. Probab. Lett., vol. 78, no. 17, pp. 2971–2975, 2008.

F. Hayashi, Econometrics. Princeton University Press, 2011.

N. K. Vishnoi, Algorithms for convex optimization. Cambridge University Press, 2021.

N. Dibal, H. Bakari, and A. Yahaya, “Estimating the parameters in the two-parameter Weibull model using simulation study and real-life data,” IOSR J. Math., vol. 12, pp. 38–42, 2016.

F. Hamad, N. Younus, M. M. Muftah, and M. Jaber, “SPECIFY UNDERLINING DISTRIBUTION FOR CLUSTERING LINEARLY SEPARABLE DATA: NORMAL AND UNIFORM DISTRIBUTION CASE,” J. Data Acquis. Process., vol. 38, no. 2, p. 4675, 2023.

P. Y. Lum et al., “Extracting insights from the shape of complex data using topology,” Sci. Rep., vol. 3, no. 1, pp. 1–8, 2013.

Published
2024-08-11
How to Cite
Alharbi , T., & Hamad , F. (2024). An Optimal Strategy for Estimating Weibull distribution Parameters by Using Maximum Likelihood Method. Statistics, Optimization & Information Computing, 12(5), 1342-1351. https://doi.org/10.19139/soic-2310-5070-2019
Issue
Section
Research Articles