Length-Biased Discretized Fréchet-Weibull Probability Distribution: Mathematical Theory, Simulation Analysis, and Goodness-of-Fit Evaluation Using Real-Life Data

Abstract

As data becomes more complex and abundant, the need for innovative statistical tools grows. This study introduces a new modeling framework based on a length-biased variant of the discretized Fr ́echet-Weibull distribution, designed to handle intricate data structures often encountered in practical applications. The paper explores the mathematical foundation of this dis-
tribution and derives its probability functions, providing an intuitive understanding of its behavior across various parameters. It highlights the flexibility of the model to capture a range of hazard rate forms, including increasing, decreasing, upside-down bathtub-shaped, and unimodal functions, making it suitable for diverse failure-time and count-data scenarios. A key feature is the model’s ability to address overdispersion and underdispersion, as well as asymmetric data behaviors. It is effective in scenarios with excess zeros, common in fields like insurance and epidemiology. The characteristics of the length-biased discretized Fr ́echet-Weibull distribution are examined using conditional expectations and reverse hazard rate functions. Parameter estimation is performed using maximum likelihood estimation, with a simulation study assessing the reliability and efficiency of the estimators. The practical value of the model is demonstrated through applications to two real datasets, showing a superior fit compared to several established alternatives.