General Form Moment-based Estimator of Weibull, Gamma, and Log-normal Distributions

Abstract: This paper presents a unified and novel estimation framework for the Weibull, Gamma, and Log-normal distributions based on arbitrary-order moment pairs. Traditional estimation techniques, such as Maximum Likelihood Estimation (MLE) and the classical Method of Moments (MoM), are often restricted to fixed-order moment inputs and may require specific distributional assumptions or optimization procedures. In contrast, our general-form moment-based estimator allows the use of any two empirical moments, such as mean and variance, or higher-order combinations, to compute the underlying distribution parameters. For each distribution, we develop provably convergent numerical algorithms that guarantee unique solutions within a bounded parameter space and provide estimates within a user-defined error tolerance. The proposed framework generalizes existing estimation methods and offers greater flexibility and robustness for statistical modeling in diverse application domains. This is, to our knowledge, the first work that formalizes such a general estimation structure and provides theoretical guarantees across these three foundational distributions.