[Paper Review] Bivariate Exponentaited Generalized Weibull-Gompertz Distribution

M. A. El-Damcese, Abdelfattah Mustafa+1 more|arXiv (Cornell University)|Jan 9, 2015

Statistical Distribution Estimation and Applications5 references21 citations

TL;DR

This paper proposes a new bivariate exponentiated generalized Weibull-Gompertz distribution based on the Marshall-Olkin family, designed for modeling dependent survival data. It derives key statistical properties including joint and marginal densities, moments, maximum likelihood estimation, and reversed hazard functions, demonstrating superior fit to real data compared to existing models.

ABSTRACT

In this paper, we introduce a bivariate exponentaited generalized Weibull-Gompertz distribution. The model introduced here is of Marshall-Olkin type. Several properties are studied such as bivariate probability density function and it is marginal, moments, maximum likelihood estimation, joint reversed (hazard) function and joint mean waiting time and it is marginal. A real data set is analyzed and it is observed that the present distribution can provide a better fit than some other very well-known distributions.

Motivation & Objective

• To develop a flexible bivariate lifetime distribution capable of modeling dependent failure times in survival analysis.
• To extend the generalized Weibull-Gompertz distribution using the Marshall-Olkin bivariate family to allow for dependence between components.
• To derive and analyze key statistical properties such as joint density, marginal distributions, moments, and reversed hazard functions.
• To evaluate the model’s performance using real data and compare it with well-known competing distributions.
• To provide a practical tool for reliability and survival analysis with enhanced modeling capability for dependent systems.

Proposed method

• The proposed distribution is constructed using the Marshall-Olkin bivariate family, which introduces dependence through a dependence parameter.
• The joint probability density function is derived by combining the generalized Weibull-Gompertz distribution with the exponentiated and bivariate extensions.
• Marginal distributions are obtained by integrating the joint density over the support of the other variable.
• Moments are derived using the moment generating function of the baseline distribution and the transformation properties of the bivariate model.
• Maximum likelihood estimation is applied to estimate model parameters using observed data.
• The joint reversed hazard function and joint mean waiting time are derived to support reliability and risk assessment applications.

Experimental results

Research questions

• RQ1 Can a bivariate extension of the generalized Weibull-Gompertz distribution provide better fit to dependent survival data than existing models?
• RQ2 How do the dependence parameters in the Marshall-Olkin framework affect the joint and marginal behavior of failure times?
• RQ3 What are the analytical properties of the proposed distribution, including moments and reversed hazard functions?
• RQ4 How does the model perform in fitting real-world survival data compared to standard distributions?
• RQ5 Can the model effectively capture both dependence and non-monotonic failure rates in reliability applications?

Key findings

• The proposed bivariate exponentiated generalized Weibull-Gompertz distribution exhibits greater flexibility in modeling dependent failure times compared to standard distributions.
• The model successfully captures non-monotonic hazard rates and provides a better fit to real data than competing models such as the bivariate Weibull and Gompertz distributions.
• Maximum likelihood estimation is feasible and yields stable parameter estimates, confirming the model's practical applicability.
• The joint reversed hazard function and joint mean waiting time are derived and shown to be useful in reliability and risk analysis.
• The marginal distributions are analytically tractable and retain the flexibility of the baseline generalized Weibull-Gompertz distribution.
• Empirical results from real data analysis confirm that the proposed model outperforms several well-known distributions in terms of goodness-of-fit criteria.