A new Weibull-Gamma distribution: theory, estimation and applications

Abstract

In the fields of reliability engineering, survival analysis, and lifetime data modeling, accurately representing the failure times and life durations of systems, components, and organisms is a central concern. Traditional lifetime distributions—such as the exponential, Weibull, and gamma distributions—have been widely used due to their mathematical tractability and interpretability. However, these classical models often struggle to capture the complexity of real-world data, particularly when hazard rate behaviors vary, including increasing, decreasing, bathtub-shaped, or unimodal patterns. Although the Weibull distribution is popular for its flexibility, it may not sufficiently model datasets where the hazard function deviates from its typical monotonic form. Similarly, while the gamma distribution is effective in many stochastic and queuing contexts, it lacks the versatility to represent certain tail behaviors and multimodal characteristics observed in practice. To address these limitations, statisticians have developed hybrid and compound distributions that merge features from multiple distributions, enhancing both flexibility and applicability. One such development is the Weibull-Gamma distribution, derived by mixing the Weibull and Gamma distributions. The first four moments about the origin, as well as the mean, were calculated for this new distribution. Derived expressions also include the coefficient of variation, skewness, kurtosis, and index of dispersion. In addition, the moment-generating function, characteristic function, and Laplace transform were established. Key reliability functions—such as the survival function, hazard rate function, and mean residual life—were also derived. Parameter estimation was carried out using the Maximum Likelihood Estimation (MLE) method. The goodness-of-fit of the proposed distribution was evaluated against several existing related models using criteria such as the Akaike Information Criterion (AIC), Corrected Akaike Information Criterion (AICC), and Bayesian Information Criterion (BIC). These comparisons were based on real-world datasets. The results demonstrated that the Weibull-Gamma distribution outperformed the competing models, making it a promising alternative for modeling real-life lifetime data.