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Mathematical Problems in Engineering
Volume 2014, Article ID 510929, 11 pages
http://dx.doi.org/10.1155/2014/510929
Research Article

Bivariate Nonlinear Diffusion Degradation Process Modeling via Copula and MCMC

1Department of Industrial Engineering, Southeast University, Nanjing 211189, China
2Department of Mathematics, Hubei Engineering University, Xiaogan 432100, China

Received 27 October 2013; Revised 12 February 2014; Accepted 14 February 2014; Published 30 March 2014

Academic Editor: Xi Frank Xu

Copyright © 2014 Huibing Hao and Chun Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. Q. Meeker and L. A. Escobar, Statistical Method for Reliability Data, John Wiley & Sons, New York, NY, USA, 1998.
  2. W. Nelson, Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, John Wiley & Sons, New York, NY, USA, 1990.
  3. M. J. Zuo, R. C. M. Yam, and K. Jiang, “Approaches for reliability modeling of continuous-strfi devices,” IEEE Transactions on Reliability, vol. 48, no. 1, pp. 9–18, 1999. View at Publisher · View at Google Scholar · View at Scopus
  4. H. W. Wang, J. Gao, and Z. Y. Liu, “Maintenance decision based on data fusion of aero engines,” Mathematical Problems in Engineering, vol. 2013, Article ID 628792, 10 pages, 2013. View at Publisher · View at Google Scholar
  5. C. Y. Li and Y. M. Zhang, “Time variant reliability assessment and its sensitivity analysis of cutting tool under invariant machining condition based on Gamma process,” Mathematical Problems in Engineering, vol. 2012, Article ID 676923, 19 pages, 2012. View at Publisher · View at Google Scholar
  6. M.-Y. Lee and J. Tang, “A modified em-algorithm for estimating the parameters of inverse gaussian distribution based on time-censored wiener degradation data,” Statistica Sinica, vol. 17, no. 3, pp. 873–893, 2007. View at Google Scholar · View at Scopus
  7. J. Tang and T.-S. Su, “Estimating failure time distribution and its parameters based on intermediate data from a wiener degradation model,” Naval Research Logistics, vol. 55, no. 3, pp. 265–276, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. Q. H. Duan, Z. P. Chen, and D. F. Zhao, “An expectation maximization algorithm to model failure times by continuous-time Markov chains,” Mathematical Problems in Engineering, vol. 2010, Article ID 242567, 26 pages, 2010. View at Publisher · View at Google Scholar · View at Scopus
  9. S.-T. Tseng, J. Tang, and I.-H. Ku, “Determination of burn-in parameters and residual life for highly reliable products,” Naval Research Logistics, vol. 50, no. 1, pp. 1–14, 2003. View at Publisher · View at Google Scholar · View at Scopus
  10. C.-Y. Peng and S.-T. Tseng, “Mis-specification analysis of linear degradation models,” IEEE Transactions on Reliability, vol. 58, no. 3, pp. 444–455, 2009. View at Publisher · View at Google Scholar · View at Scopus
  11. C. Park and W. J. Padgett, “Accelerated degradation models for failure based on geometric Brownian motion and gamma processes,” Lifetime Data Analysis, vol. 11, no. 4, pp. 511–527, 2005. View at Publisher · View at Google Scholar · View at Scopus
  12. N. Z. Gebraeel, M. A. Lawley, R. Li, and J. K. Ryan, “Residual-life distributions from component degradation signals: a Bayesian approach,” IIE Transactions, vol. 37, no. 6, pp. 543–557, 2005. View at Publisher · View at Google Scholar · View at Scopus
  13. X. Wang, “Wiener processes with random effects for degradation data,” Journal of Multivariate Analysis, vol. 101, no. 2, pp. 340–351, 2010. View at Publisher · View at Google Scholar · View at Scopus
  14. K. A. Doksum and A. Hoyland, “Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution,” Technometrics, vol. 34, no. 1, pp. 74–82, 1992. View at Google Scholar · View at Scopus
  15. G. A. Whitmore and F. Schenkelberg, “Modeling accelerated degradation data using Wiener diffusion with a time scale transformation,” Lifetime Data Analysis, vol. 3, no. 1, pp. 27–45, 1997. View at Google Scholar · View at Scopus
  16. J. Fan, K. C. Yung, and M. Pecht, “Physics-of-failure-based prognostics and health management for high-power white light-emitting diode lighting,” IEEE Transactions on Device and Materials Reliability, vol. 11, no. 3, pp. 407–416, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. V. Crk, “Reliability assessment from degradation data,” in Proceedings of the Annual Reliability and Maintainability Symposium, pp. 155–161, Los Angeles, Calif, USA, January 2000. View at Scopus
  18. W. Huang and R. G. Askin, “Reliability analysis of electronic devices with multiple competing failure modes involving performance aging degradation,” Quality and Reliability Engineering International, vol. 19, no. 3, pp. 241–254, 2003. View at Publisher · View at Google Scholar · View at Scopus
  19. V. Bagdonavičius, A. Bikelis, and V. Kazakevičius, “Statistical analysis of linear degradation and failure time data with multiple failure modes,” Lifetime Data Analysis, vol. 10, no. 1, pp. 65–81, 2004. View at Publisher · View at Google Scholar · View at Scopus
  20. V. Bagdonavičius, A. Bikelis, V. Kazakevičius, and M. Nikulin, “Analysis of joint multiple failure mode and linear degradation data with renewals,” Journal of Statistical Planning and Inference, vol. 137, no. 7, pp. 2191–2207, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. J. K. Sari, Multivariate degradation modeling and its application to reliability testing [Ph.D. thesis], National University of Singapore, Singapore, 2007.
  22. J. K. Sari, M. J. Newby, A. C. Brombacher, and L. C. Tang, “Bivariate constant stress degradation model: led lighting system reliability estimation with two-stage modelling,” Quality and Reliability Engineering International, vol. 25, no. 8, pp. 1067–1084, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. Z. Pan, N. Balakrishnan, and Q. Sun, “Bivariate constant-stress accelerated degradation model and inference,” Communications in Statistics, vol. 40, no. 2, pp. 259–269, 2011. View at Publisher · View at Google Scholar · View at Scopus
  24. Z. Q. Pan, N. Balakrishnan, Q. Sun, and J. L. Zhou, “Bivariate degradation analysis of products based on Wiener processes and copulas,” Communications in Statistics, vol. 83, pp. 1316–1329, 2013. View at Google Scholar
  25. I. Ntzoufras, Bayesian Modeling Using WinBUGS, John Wiley & Sons, New York, NY, USA, 2009.
  26. W. K. Hastings, “Monte carlo sampling methods using Markov chains and their applications,” Biometrika, vol. 57, no. 1, pp. 97–109, 1970. View at Publisher · View at Google Scholar · View at Scopus
  27. A. Smith and G. Roberrs, “Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods,” Journal of the Royal Statistical Society B, vol. 55, pp. 3–23, 1993. View at Google Scholar
  28. R. B. Nelson, An Introduction to Copulas, Springer Science, New York, NY, USA, 2006.
  29. V. N. H. Chaluvadi, Accelerated life testing of electronic revenue meters [Ph.D. thesis], Clemson University, 2008.
  30. J. M. van Noortwijk, “A survey of the application of gamma processes in maintenance,” Reliability Engineering and System Safety, vol. 94, no. 1, pp. 2–21, 2009. View at Publisher · View at Google Scholar · View at Scopus
  31. G. Y. Liu and M. Lu, “The distribution of extreme value for Brownian motion with nonlinear drift,” Journal of Mathematics, vol. 30, pp. 315–319, 2010. View at Google Scholar
  32. X. S. Si, W. Wang, and D. H. Zhou, “A generalized result for degradation model-based reliability estimation,” IEEE Transactions on Automation Science and Engineering, 2013. View at Google Scholar