On the Maximization of the Likelihood for the Generalized Gamma Distribution: the Modified Maximum Likelihood Approach

Abstract

Maximum likelihood (ML) estimation of parameters of the generalized gamma (GG) distribution has been considered in several papers, and some of them stated that the ML estimation has some computational difficulties. Therefore, different approaches including numerical methods have been proposed for the ML estimation of parameters of the GG distribution. However, it is known that using numerical methods may have some drawbacks, e.g., non-convergence of iterations, multiple roots, and convergence to the wrong root. In this study, we rehabilitate the ML procedure via the modified ML (MML) methodology and obtain the likelihood equations in which two of them have explicit solutions, and the remaining one should be solved numerically. Since the MML methodology explicitly solves two of three likelihood equations, the mentioned drawbacks are alleviated. We also propose a simple algorithm to obtain the estimates of the parameters of the GG distribution. Then, the GG distribution is used for modeling the real data sets, and the performance of the proposed algorithm is compared with the Broyden–Fletcher–Goldfarby–Shanno (BFGS) and Nelder–Mead (NM) algorithms. The results show that the proposed algorithm is preferable to the BFGS and NM algorithms in terms of computational sense when considering the GG distribution. © The Author(s) 2025.