[Paper Review] The Generalized Power Generalized Weibull Distribution: Properties and Applications

Mahmoud Ali Selim|arXiv (Cornell University)|Jul 27, 2018

Statistical Distribution Estimation and Applications14 references4 citations

TL;DR

This paper proposes the generalized power generalized Weibull (GPGW) distribution, a flexible statistical model that extends the Weibull and power generalized Weibull distributions. It offers a wide range of hazard rate shapes—including bathtub and upside-down bathtub—enabling improved modeling of complex failure time data. The key contribution lies in its comprehensive derivation of statistical properties and successful application to real-world data using maximum likelihood estimation.

ABSTRACT

This paper introduces a new generalization of the power generalized Weibull distribution called the generalized power generalized Weibull distribution. This distribution can also be considered as a generalization of Weibull distribution. The hazard rate function of the new model has nice and flexible properties and it can take various shapes, including increasing, decreasing, upside-down bathtub and bathtub shapes. Some of the statistical properties of the new model, including quantile function, moment generating function, reliability function, hazard function and the reverse hazard function are obtained. The moments, incomplete moments, mean deviations and Bonferroni and Lorenz curves and the order statistics densities are also derived. The model parameters are estimated by the maximum likelihood method. The usefulness of the proposed model is illustrated by using two applications of real-life data.

Motivation & Objective

• To develop a new flexible statistical distribution that generalizes the Weibull and power generalized Weibull distributions.
• To provide a model with versatile hazard rate functions capable of capturing increasing, decreasing, bathtub, and upside-down bathtub patterns.
• To derive comprehensive statistical properties including moments, reliability, and hazard functions for enhanced modeling capability.
• To estimate model parameters using maximum likelihood estimation for practical applicability.
• To demonstrate the model's effectiveness through real data applications in reliability and survival analysis.

Proposed method

• The GPGW distribution is constructed by introducing additional shape parameters to the standard Weibull distribution, enhancing its flexibility.
• The hazard rate function is derived and analyzed to show its ability to assume various shapes, including bathtub and upside-down bathtub.
• Statistical properties such as the quantile function, moment generating function, reliability function, and reverse hazard function are mathematically derived.
• Moments, incomplete moments, mean deviations, and Bonferroni and Lorenz curves are derived to support risk and inequality analysis.
• Order statistics densities are obtained to support extreme value and record data modeling.
• Parameter estimation is performed via the maximum likelihood method, with numerical optimization used in the real data applications.

Experimental results

Research questions

• RQ1 Can a new generalized Weibull distribution be developed that offers greater flexibility in modeling diverse hazard rate shapes?
• RQ2 How do the statistical properties of the GPGW distribution compare to those of existing Weibull extensions?
• RQ3 To what extent can the GPGW model outperform standard Weibull and related distributions in fitting real-life failure time data?
• RQ4 What are the analytical forms of key reliability and risk measures (e.g., moments, Bonferroni curve) in the GPGW framework?
• RQ5 How robust and efficient is maximum likelihood estimation for the parameters of the GPGW distribution in practical settings?

Key findings

• The generalized power generalized Weibull distribution exhibits flexible hazard rate functions capable of modeling increasing, decreasing, bathtub, and upside-down bathtub failure rates.
• The derived moments, incomplete moments, and mean deviations provide a solid foundation for reliability and risk assessment in survival analysis.
• Bonferroni and Lorenz curves are explicitly derived, enabling inequality and income distribution analysis in applied contexts.
• The model demonstrates superior fit to two real-life data sets compared to baseline Weibull and power generalized Weibull distributions.
• Maximum likelihood estimation of the parameters is feasible and effective, as confirmed by the successful application to empirical data with convergence in numerical optimization.