Parameter Estimation of Generalized Modified Weibull Using the Maximum Likelihood on Simulation and Real-World Data

Muhammad Luthfi Setiarno Putera; Purhadi Purhadi

doi:10.37905/jjom.v8i1.36518

Parameter Estimation of Generalized Modified Weibull Using the Maximum Likelihood on Simulation and Real-World Data

Muhammad Luthfi Setiarno Putera, Purhadi Purhadi

Abstract

This study estimates parameters of the generalized modified Weibull (GM Weibull) distribution using the Maximum Likelihood Estimation (MLE) method with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. The GM Weibull distribution, which includes four parameters (lambda, theta, phi, tau), offers greater flexibility than Weibull distribution in modeling data with monotonic and bathtub-shaped hazard patterns. Parameter estimation was conducted on three datasets: simulated data with sample sizes of 50, 200, and 500 observations; survival data from 45 heart transplant patients; and health indicator data from 27 districts/cities in Central and South Kalimantan provinces. The results demonstrate that while the standard Weibull remains a parsimonious choice for simple monotonic data, the GM Weibull produces parameter estimates closer to theoretical values in small-to-medium samples and significantly lower deviance in complex datasets. Specifically, for the heart transplant data, the GM Weibull offered better modeling long-term survival tails (800--1,000 days), while for the health indicator data, it effectively accommodated central tendencies within asymmetric distributions. Although AIC and BIC favor standard Weibull, the GM Weibull accurately identifies underlying structural fluctuations and non-monotonic failure characteristics. This study confirms that the MLE-based GM Weibull distribution is one of the robust tools for researchers requiring a more representative model for complex survival and health data.

Keywords

BFGS; Generalized Modified Weibull; Parameter Estimation; Maximum Likelihood

Full Text:

PDF

References

M. Shahriari, H. Shahrasbi, and A. Zaretalab, “Reliability analysis of lifetime systems based on Weibull distribution,” Int. J. Nonlinear Anal. Appl., vol. 15, pp. 2008–6822, 2024, doi: 10.22075/ijnaa.2024.31309.4609.

T. Cavalcante, R. Ospina, V. Leiva, X. Cabezas, and C. Martin-Barreiro, “Weibull regression and machine learning survival models: Methodology, comparison, and application to biomedical data related to cardiac surgery,” Biology, vol. 12, no. 3, p. 442, Mar. 2023, doi: 10.3390/biology12030442.

I. Wulandari, A. Kurnia, and K. Sadik, “Weibull regression and stratified Cox regression in modelling exclusive breastfeeding duration,” J. Phys. Conf. Ser., vol. 1940, no. 1, p. 012001, Jun. 2021, doi: 10.1088/1742-6596/1940/1/012001.

M. S. Shama et al., “Modified generalized Weibull distribution: Theory and applications,” Sci. Rep., vol. 13, no. 1, pp. 1–16, Aug. 2023, doi: 10.1038/s41598-023-38942-9.

D. Ozonur, I. Pobocikova, and A. de Souza, “Statistical analysis of monthly rainfall in Central West Brazil using probability distributions,” Model. Earth Syst. Environ., vol. 7, no. 3, pp. 1979–1989, Sep. 2021, doi: 10.1007/s40808-020-00954-z.

J. Mazucheli, A. F. B. Menezes, L. B. Fernandes, R. P. de Oliveira, and M. E. Ghitany, “The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates,” J. Appl. Stat., vol. 47, no. 6, p. 954, Apr. 2019, doi: 10.1080/02664763.2019.1657813.

H. Wang, B. Abba, and J. Pan, “Classical and Bayesian estimations of improved Weibull–Weibull distribution for complete and censored failure times data,” Appl. Stoch. Models Bus. Ind., vol. 38, no. 6, pp. 997–1018, Nov. 2022, doi: 10.1002/asmb.2698.

N. Taketomi, K. Yamamoto, C. Chesneau, and T. Emura, “Parametric distributions for survival and reliability analyses: A review and historical sketch,” Mathematics, vol. 10, no. 20, p. 3907, Oct. 2022, doi: 10.3390/math10203907.

S. Buzrul, “The Weibull model for microbial inactivation,” Food Eng. Rev., vol. 14, no. 1, pp. 45–61, Mar. 2022, doi: 10.1007/s12393-021-09291-y.

M. Imran et al., “The development of an extended Weibull model with applications to medicine, industry and actuarial sciences,” Sci. Rep., vol. 14, no. 1, p. 12338, May 2024, doi: 10.1038/s41598-024-61308-8.

X. Yang, L. Xie, B. Zhao, X. Kong, and N. Wu, “An iterative method for parameter estimation of the three-parameter Weibull distribution based on a small sample size with a fixed shape parameter,” Int. J. Inf. Technol. Decis. Mak., vol. 22, no. 12, Apr. 2022, doi: 10.1142/S0219455422501255.

Y. Wu, H. Xie, J. Y. Chiang, G. Peng, and Y. Qin, “Parameter estimation and applications of the Weibull distribution for strength data of glass fiber,” Math. Probl. Eng., vol. 2021, no. 1, p. 9175170, Jan. 2021, doi: 10.1155/2021/9175170.

J. M. F. Carrasco, E. M. M. Ortega, and G. M. Cordeiro, “A generalized modified Weibull distribution for lifetime modeling,” Comput. Stat. Data Anal., vol. 53, no. 2, pp. 450–462, Dec. 2008, doi: 10.1016/j.csda.2008.08.023.

M. E. Omer, M. A. Bakar, M. Adam, and M. Mustafa, “Utilization of a mixture cure rate model based on the generalized modified Weibull distribution for the analysis of leukemia patients,” Asian Pac. J. Cancer Prev., vol. 22, no. 4, p. 1045, Apr. 2021, doi: 10.31557/APJCP.2021.22.4.1045.

C.-D. Lai, Generalized Weibull Distributions. Berlin, Heidelberg: Springer, 2014, doi: 10.1007/978-3-642-39106-4.

S. M. A. Aljeddani and M. A. Mohammed, “Parameter estimation of a model using maximum likelihood function and Bayesian analysis through moment of order statistics,” Alexandria Eng. J., vol. 75, pp. 221–232, Jul. 2023, doi: 10.1016/j.aej.2023.05.079.

M. E. Prasetya, P., and S., “Parameter estimation and hypothesis testing of geographically and temporally weighted bivariate Weibull regression,” KnE Life Sci., pp. 239–252, Mar. 2024, doi: 10.18502/kls.v8i1.15586.

E. K. Nisa and S. D. Miasary, “Parameter estimation and application of inverse Gaussian regression,” AIP Conf. Proc., vol. 3046, no. 1, Feb. 2024, doi: 10.1063/5.0194666.

A. Mehrabi, O. Altintas, and J. W. Morphew, “Optimizing maximum likelihood estimation in performance factor analysis: A comparative study of estimation methods,” Springer Proc. Math. Stat., vol. 452, pp. 223–232, 2024, doi: 10.1007/978-3-031-55548-0_21.

J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data. Hoboken, NJ, USA: John Wiley & Sons, 2002, doi: 10.1002/9781118032985.

M. Korkmaz and C. Chesneau, “On the unit Burr-XII distribution with the quantile regression modeling and applications,” Comput. Appl. Math., vol. 40, no. 1, p. 29, Feb. 2021, doi: 10.1007/s40314-021-01418-5.

H. H. Ahmad and K. A. Elnagar, “A novel quantile regression for fractiles based on unit logistic exponential distribution,” AIMS Math., vol. 9, no. 12, pp. 34504–34536, 2024, doi: 10.3934/math.20241644.

Badan Pusat Statistik Provinsi Kalimantan Tengah, “Distribusi persentase penduduk yang mempunyai keluhan kesehatan selama sebulan terakhir dan tidak berobat jalan menurut kabupaten/kota dan alasan utama tidak berobat jalan di Provinsi Kalimantan Tengah, 2023.” [Online]. Available: https://kalteng.bps.go.id. Accessed: Dec. 31, 2025.

Badan Pusat Statistik Provinsi Kalimantan Selatan, “Distribusi persentase penduduk yang mempunyai keluhan kesehatan selama sebulan terakhir dan tidak berobat jalan menurut kabupaten/kota dan alasan utama tidak berobat jalan di Provinsi Kalimantan Selatan, 2023.” [Online]. Available: https://kalsel.bps.go.id. Accessed: Dec. 31, 2025.

G. M. Cordeiro, M. H. Tahir, J. C. S. Vasconcelos, E. M. M. Ortega, and M. A. Hussain, “A new extended log-Weibull regression: Simulations and applications,” Hacettepe J. Math. Stat., vol. 50, no. 3, pp. 855–871, Jun. 2021, doi: 10.15672/hujms.779454.

M. L. S. Putera, L. Wahyunita, and F. Yusup, “Spatial modelling of Covid-19 confirmed cases in Kalimantan, Indonesia: How neighborhood matters?,” Walailak J. Sci. Technol., vol. 18, no. 15, p. 22120, Jul. 2021, doi: 10.48048/wjst.2021.22120.

C. Agiakloglou and A. Tsimpanos, “Evaluating the performance of AIC and BIC for selecting spatial econometric models,” J. Spat. Econom., vol. 4, no. 2, 2022, doi: 10.1007/s43071-022-00030-x.

M. Alizadeh, M. N. Khan, M. Rasekhic, and G. G. Hamedani, “A new generalized modified Weibull distribution,” Stat. Optim. Inf. Comput., vol. 9, no. 1, pp. 17–34, Jan. 2021, doi: 10.19139/soic-2310-5070-1014.

DOI: https://doi.org/10.37905/jjom.v8i1.36518