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

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.