Maximum Likelihood Estimation for the Log-Logistic Distribution Using Whale Optimization Algorithm with Applications

Öz

The log-logistic distribution has been widely used in several fields, including engineering, survival analysis, and economics. The method of maximum likelihood estimation is used in this study for estimating the shape and scale parameters for the log-logistic distribution, whereas in the case of the log-logistic distribution, likelihood equations lack explicit solutions. Therefore, problems with solving likelihood equations can be solved by using two highly efficient algorithms, which are the whale optimization algorithm and the Nelder-Mead algorithm, as well as by showing the applicability of this distribution by comparing it with other well-known classical distributions. To demonstrate the performance of each algorithm implemented, an extensive Monte Carlo simulation study has been conducted. The performance of maximum likelihood estimators for each algorithm has been evaluated in terms of mean square error and deficiency criteria. It has been seen that the whale optimization algorithm provides the best estimates for the log-logistic distribution parameters according to the simulation data.

Anahtar Kelimeler

• Maximum likelihood
• Log-logistic distribution
• Whale optimization
• Monte-Carlo simulation

Kaynakça

1. Abbas K, Tang Y. 2016. Objective Bayesian analysis for log-logistic distribution. Commun Stat-Simul Comput, 45(8): 2782-2791. DOI: 10.1080/03610918.2014.925925.
2. Aldeni M, Lee C, Famoye F. 2017. Families of distributions arising from the quantile of generalized lambda distribution. J Stat Distribut Appl, 4: 1-18. DOI: 10.1186/s40488-017-0081-4.
3. Ali M, Khan A. 1987. On order statistics from the log-logistic distribution. J Stat Plan Infer, 17: 103-108. DOı: 10.1016/0378-3758(87)90104-2.
4. Al-Mhairat B, Al-Quraan A. 2022. Assessment of wind energy resources in Jordan using different optimization techniques. Processes, 10(1): 105. DOI: 10.3390/pr10010105.
5. Anderson D, Burnham K, White G. 1998. Comparison of Akaike information criterion and consistent Akaike information criterion for model selection and statistical inference from capture-recapture studies. J Appl Stat, 25(2): 263-282. DOI: 10.1080/02664769823250.
6. Balakrishnan N, Malik H. 1987. Moments of order statistics from truncated log-logistic distribution. J Stat Plann Infer, 17: 251-267. DOI: 10.1016/0378-3758(87)90117-0.
7. Burr I. 1942. Cumulative frequency functions. Annals Math Stat, 13(2): 215-232. DOI: 10.1214/aoms/1177731607.
8. Everitt B. 1984. Maximum likelihood estimation of the parameters in a mixture of two univariate normal distributions; a comparison of different algorithms. J Royal Stat Soc Series D: The Statistician, 33(2): 205-215. DOI: 10.2307/2987851.

9. Fisk P. 1961. The graduation of income distributions. Econometrica, 1961: 171-185. DOI: 10.2307/1909287.
10. Gao F, Han L. 2012. Implementing the Nelder-Mead simplex algorithm with adaptive parameters. Comput Optimiz Appl, 51(1): 259-277. DOI: 10.1007/s10589-010-9329-3.
11. Gross A, Clark V. 1975. Survival distributions: Reliability applications in the biometrical sciences. John Wiley, New York, US.
12. Hu H, Bai Y, Xu T. 2016. A whale optimization algorithm with inertia weight. WSEAS Trans Comput, 15: 319-326.
13. Ijaz M, Asim S, Alamgir, Farooq M, Khan S, Manzoor S. 2020. A Gull Alpha Power Weibull distribution with applications to real and simulated data. Plos One, 15(6): e0233080. DOI: 10.1371/journal.pone.0233080.
14. Kantam R, Srinivasa R. 2002. Log-logistic distribution: modified maximum likelihood estimation. Gujarat Stat Rev, 29(1): 25-36.
15. Kucukdeniz T, Esnaf S. 2018. Hybrid revised weighted fuzzy c-means clustering with Nelder-Mead simplex algorithm for generalized multisource Weber problem. J Enterp Info Manag, 31(6): 908-924. DOI: 10.1108/JEIM-01-2018-0002.
16. Kus C, Kaya M. 2006. Estimation of parameters of the loglogistic distribution based on progressive censoring using the EM algorithm. Hacettepe J Math Stat, 35(2): 203-211.
17. Lee E, Wang J. 2003. Statistical methods for survival data analysis, John Wiley & Sons, New York, US, pp: 476.
18. Lemonte A, Cordeiro G. 2011. An extended Lomax distribution. Statistics, 47(4): 800-816. DOI: 10.1080/02331888.2011.568119.
19. Marthin P, Rao G. 2020. Generalized Weibull–Lindley (GWL) distribution in modeling lifetime data. J Math, 2020: 2049501. DOI: 10.1155/2020/2049501.
20. Mirjalili S, Lewis A. 2016. The whale optimization algorithm. Adv Eng Software, 95: 51-67. DOI: 10.1016/j.advengsoft.2016.01.008.
21. Mohammed W, Elmasry W. 2023. A comparative assessment of five different distributions based on five different optimization methods for modeling wind speed distribution. Gazi Univ J Sci, 2023: 1-1. DOI: 10.35378/gujs.1026834.
22. Mohammed W. 2021. Five different distributions and metaheuristics to model wind speed distribution. J Thermal Eng, 7(14): 1898-1920. DOI: 10.14744/jten.2021.xxxx.
23. Nelder J, Mead R. 1965. A simplex method for function minimization. Comput J, 7(4): 308-313. DOI: 10.1093/comjnl/7.4.308.
24. Pratihar D. 2012. Traditional vs non-traditional optimization tools. In: Basu K (ed) Computational Optimization and Applications, Narosa Publishing House Pvt. Ltd, New Delhi, India, pp: 25-33.
25. Rana N, Latiff M, Abdulhamid S, Chiroma H. 2020. Whale optimization algorithm: a systematic review of contemporary applications, modifications and developments. Neural Comput Appl, 32(20): 16245-16277. DOI: 10.1007/s00521-020-04849-z.
26. Shah B, Dave P. 1963. A note on log-logistic distribution. J MS Univ Baroda, 12: 15-20.
27. Shamir R. 1987. The efficiency of the simplex method: a survey. Manag Sci, 33(3): 301-334. DOI: 10.1287/mnsc.33.3.301.
28. Shanker R, Shukla K, Shanker R, Leonida T. 2016. On modeling of lifetime data using two-parameter Gamma and Weibull distributions. Biomet Biostat Int J, 4(5): 201-206. DOI: 10.15406/bbij.2016.04.00107.
29. Shoukri M, Mian I, Tracy D. 1988. Sampling properties of estimators of the log‐logistic distribution with application to Canadian precipitation data. Canadian J Stat, 16(3): 223-236. DOI: 10.2307/3314729.
30. Shukla K. 2019. A comparative study of one parameter lifetime distributions. Biomet Biostat Int J, 8(4): 111-123. DOI: 10.15406/bbij.2019.08.00280.
31. Sreenivas P, Kumar S. 2015. A review on non-traditional optimization algorithm for simultaneous scheduling problems. J Mech Civil Eng, 12(2): 50-53. DOI: 10.9790/1684-12225053.
32. Tadikamalla P, Johnson N. 1982. Systems of frequency curves generated by transformations of logistic variables. Biometrika, 69(2): 461-465. DOI: 10.1093/biomet/69.2.461.
33. Tadikamalla P. 1980. A look at the Burr and related distributions. Int Stat Rev, 337-344. DOI: 10.2307/1402945.
34. Yan Z, Wang S, Liu B, Li X. 2018. Application of whale optimization algorithm in optimal allocation of water resources. E3S Web of Conferences, 53: 04019. EDP Sciences. DOI: 10.1051/e3sconf/20185304019.
35. Yuan K, Schuster C. 2013. Overview of statistical estimation methods. The Oxford handbook of quantitative methods. Oxford, UK. DOI: 10.1093/oxfordhb/9780199934874.013.0018.
36. Zea L, Silva R, Bourguignon M, Santos A, Cordeiro G. 2012. The beta exponentiated Pareto distribution with application to bladder cancer susceptibility. Int J Stat Probab, 1(2): 8. DOI: 10.5539/ijsp.v1n2p8.