Review Articles

Step-stress accelerated degradation test planning based on Wiener process with correlation

Lei He ,

Department of Mathematics, Shanghai Normal University, Shanghai, People's Republic of China

Rong-Xian Yue ,

Department of Mathematics, Shanghai Normal University, Shanghai, People's Republic of China; Scientific Computing Key Laboratory of Shanghai Universities, Shanghai, People's Republic of China

Daojiang He

Department of Statistics, Anhui Normal University, Wuhu, People's Republic of China

Pages 58-67 | Received 26 Oct. 2017, Accepted 14 Apr. 2018, Published online: 18 May. 2018,
  • Abstract
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To assess the lifetime distribution of highly reliable or expensive product, one of the most commonly used strategies is to construct step-stress accelerated degradation test (SSADT) which can curtail the test duration and reduce the test cost. In reality, it is not unusual for a unit with a higher degradation rate which exhibits a more volatile degradation path. Recently, Ye, Chen, and Shen [(2015). A new class of Wiener process models for degradation analysis. Reliability Engineering and System Safety, 139, 58–67] proposed a Wiener process to capture the positive correlation between the drift rate and the volatility. In this paper, an optimal SSADT plan is developed under the assumption that the underlying degradation path follows the Wiener process with correlation. Firstly, the stochastic diffusion process is introduced to model a typical SSADT problem. Then the design variables, including the sample size, the measurement frequency and the numbers of measurements under each stress level, are optimised by minimising the asymptotic variance of the estimated p-percentile of the product's lifetime distribution subject to the total experimental cost not exceeding a pre-specified budget. Finally, a numerical example is presented to illustrate the proposed method.


  1. Doksum, K. A., & Normand, S. L. T. (1995). Gaussian models for degradation processes – Part I: Methods for the analysis of biomarker data. Lifetime Data Analysis, 1(2), 131144. doi: 10.1007/BF00985763 [Google Scholar]
  2. Guan, Q., Tang, Y. C., & Xu, A. C. (2015). Objective Bayesian analysis accelerated degradation test based on Wiener process models. Applied Mathematical Modelling, 40(4), 27432755. doi: 10.1016/j.apm.2015.09.076 [Google Scholar]
  3. Guida, M., Postiglione, F., & Pulcini, G. (2018). A Bayesian approach for non-homogeneous gamma degradation processes. Communications in Statistics – Theory and Methods. doi: 10.1080/03610926.2018.1440306 [Taylor & Francis Online], [Google Scholar]
  4. Hamada, M. S. (2015). Bayesian analysis of step-stress accelerated life tests and its use in planning. Quality Engineering, 27(3), 276282. doi: 10.1080/08982112.2015.1038357 [Taylor & Francis Online], [Google Scholar]
  5. Hamada, M. S., Wilson, A., & Reese, C. S. (2008). Bayesian reliability. Berlin: Springer[Google Scholar]
  6. Hu, C. H., Lee, M. Y., & Tang, J. (2015). Optimum step-stress accelerated degradation test for Wiener degradation process under constraints. European Journal of Operational Research, 241, 412421. doi: 10.1016/j.ejor.2014.09.003 [Google Scholar]
  7. Kim, Y. S., & Sung, S.-I. (2017). Practical lifetime estimation strategy based on partially step-stress-accelerated degradation tests. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 231(5), 605614[Google Scholar]
  8. Li, X. Y., Mohammad, R., Ge, Z. Z., Mohamed, A., & Lee, J. (2015). Bayesian optimal design of step stress accelerated degradation testing. Journal of Systems Engineering and Electronics, 26(3), 502513. doi: 10.1109/JSEE.2015.00058 [Google Scholar]
  9. Liao, C. M., & Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation tests. IEEE Transactions on Reliability, 55(1), 5966. doi: 10.1109/TR.2005.863811 [Google Scholar]
  10. Lim, H., & Yum, B.-J. (2011). Optimal design of accelerated degradation tests based on Wiener process models. Journal of Applied Statistics, 38(2), 309325. doi: 10.1080/02664760903406488 [Taylor & Francis Online], [Google Scholar]
  11. Lim, H.-S. (2012). Optimal design of accelerated degradation tests under the constraint of total experimental cost in the case that the degradation characteristic follows a Wiener process. Journal of the Korean Society for Quality Management, 40(2), 117125. doi: 10.7469/JKSQM.2012.40.2.117 [Google Scholar]
  12. Meeker, W. Q., & Escobar, L. A. (1998). Statistical methods for reliability data. New York, NY: Wiley[Google Scholar]
  13. Nelson, W. (1990). Accelerated testing: Statistical models, test plans, and data analyses. New York, NY: Wiley[Google Scholar]
  14. Padgett, W. J., & Tomlinson, M. A. (2004). Inference from accelerated degradation and failure data based on Gaussian process models. Lifetime Data Analysis, 10(2), 191206. doi: 10.1023/B:LIDA.0000030203.49001.b6 [Google Scholar]
  15. Pan, Z., & Balakrishnan, N. (2010). Multiple-steps step-stress accelerated degradation modeling based on Wiener and gamma processes. Communications in Statistics – Simulation and Computation, 39(39), 13841402. doi: 10.1080/03610918.2010.496060 [Taylor & Francis Online], [Google Scholar]
  16. Pan, Z., & Sun, Q. (2014). Optimal design for step-stress accelerated degradation test with multiple performance characteristics based on gamma processes. Communications in Statistics – Simulation and Computation, 43(2), 298314. doi: 10.1080/03610918.2012.700749 [Taylor & Francis Online], [Google Scholar]
  17. Park, C., & Padgett, W. J. (2006). Stochastic degradation models with several accelerating variables. IEEE Transactions on Reliability, 55(2), 379390. doi: 10.1109/TR.2006.874937 [Google Scholar]
  18. Sung, S.-I., & Yum, B.-J. (2016). Optimal design of step-stress accelerated degradation tests based on the Wiener degradation process. Quality Technology & Quantitative Management, 13(4), 367393. doi: 10.1080/16843703.2016.1189179 [Taylor & Francis Online], [Google Scholar]
  19. Tseng, S. T., Balakrishnan, N., & Tsai, C. C. (2009). Optimal step-stress accelerated degradation test plan for gamma degradation processes. IEEE Transactions on Reliability, 58(4), 611618. doi: 10.1109/TR.2009.2033734 [Google Scholar]
  20. Tseng, S.-T., & Wen, Z.-C. (2000). Step-stress accelerated degradation analysis of highly-reliable products. Journal of Quality Technology, 32, 209216. doi: 10.1080/00224065.2000.11979997 [Taylor & Francis Online], [Google Scholar]
  21. Wang, H., Wang, G. J., & Duan, F. J. (2016). Planning of step-stress accelerated degradation test based on the inverse Gaussian process. Reliability Engineering and System Safety, 154, 97105. doi: 10.1016/j.ress.2016.05.018 [Google Scholar]
  22. Whitmore, G. A., & Schenkelberg, F. (1997). Modelling accelerated degradation data using Wiener diffusion with a time scale transformation. Lifetime Data Analysis, 3(1), 2745. doi: 10.1023/A:1009664101413 [Google Scholar]
  23. Xu, A. C., & Tang, Y. C. (2015). Reference optimality criterion for planning accelerated life testing. Journal of Statistical Planning and Inference, 167, 1426. doi: 10.1016/j.jspi.2015.06.002 [Google Scholar]
  24. Ye, Z. S., Chen, L. P., Tang, L. C., & Xie, M. (2014). Accelerated degradation test planning using the inverse Gaussian process. IEEE Transactions on Reliability, 63(3), 750763. doi: 10.1109/TR.2014.2315773 [Google Scholar]
  25. Ye, Z. S., & Chen, N. (2014). The inverse Gaussian process as a degradation model. Technometrics, 56(3), 302311. doi: 10.1080/00401706.2013.830074 [Taylor & Francis Online], [Google Scholar]
  26. Ye, Z. S., Chen, N., & Shen, Y. (2015). A new class of Wiener process models for degradation analysis. Reliability Engineering and System Safety, 139, 5867. doi: 10.1016/j.ress.2015.02.005 [Google Scholar]