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Statistical Analysis for Competing Risks' Model with Two Dependent Failure Modes from Marshall-Olkin Bivariate

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Summary

This study models dependent competing failure modes using the Gompertz and Marshall-Olkin bivariate distributions. It presents both frequentist and Bayesian estimation methods with confidence intervals for reliability analysis.

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Area of Science:

  • Reliability Engineering
  • Statistical Modeling
  • Survival Analysis

Background:

  • Understanding dependent failure modes is crucial for accurate system reliability assessment.
  • Traditional models often assume independence, which can lead to underestimation of risk.
  • Competing risks introduce complexity in analyzing failure data.

Purpose of the Study:

  • To develop statistical methods for analyzing two dependent competing failure modes.
  • To apply the Gompertz distribution and Marshall-Olkin bivariate distribution for modeling dependence.
  • To derive and compare frequentist and Bayesian estimators and confidence intervals.

Main Methods:

  • Utilizing the Marshall-Olkin bivariate distribution to model the dependence structure.
  • Obtaining Maximum Likelihood Estimates (MLEs) and bootstrap confidence intervals (CIs).
  • Deriving Bayesian estimates using conjugate, Jeffreys, and Reference priors, including Highest Posterior Density (HPD) CIs.

Main Results:

  • Explicit forms for Bayesian estimates were obtained.
  • Associated bootstrap and HPD confidence intervals were constructed.
  • Numerical illustrations demonstrated the performance of the proposed methods.

Conclusions:

  • The proposed methods effectively handle dependent competing failure modes from Gompertz distributions.
  • Both frequentist and Bayesian approaches provide valuable tools for reliability analysis.
  • The study offers a robust framework for assessing systems with complex failure dependencies.