Planning step-stress test plans under Type-I hybrid censoring for the log-location-scale distribution

  • Chien-Tai LinEmail author
  • Cheng-Chieh Chou
  • N. Balakrishnan
Original Paper


The optimal design of a k-level step-stress accelerated life-testing (ALT) experiment with unequal duration steps under Type-I hybrid censoring scheme for a general log-location-scale lifetime distribution is discussed here. Censoring is allowed only at the change-stress point in the final stage. Based on the cumulative exposure model, the determination of the optimal choice for Weibull, lognormal and log-logistic lifetime distributions are considered by minimization of the asymptotic variance of the maximum likelihood estimate of the pth percentile of the lifetime at the normal operating condition. Numerical results show that for these lifetime distributions, the optimal k-step-stress ALT design with unequal duration steps under Type-I hybrid censoring scheme reduces just to a 2-step-stress ALT design.


Accelerated life-test Cumulative exposure model Fisher information matrix Maximum likelihood method 



The authors express their sincere thanks to the Editor, Associate Editor, and an anonymous reviewer for their valuable comments and suggestions on an earlier version of this manuscript which led to this improved version. The first author thanks the Ministry of Science and Technology of the Republic of China, Taiwan (Contract No: MOST 106-2118-M-032-006-MY2) for funding this research.


  1. Alhadeed AA, Yang SS (2005) Optimal simple step-stress plan for cumulative exposure model using log-normal distribution. IEEE Trans Reliab 54:64–68CrossRefGoogle Scholar
  2. Bai DS, Kim MS (1993) Optimum simple step-stress accelerated life tests for Weibull distribution and Type I censoring. Naval Res Logist 40:193–210CrossRefzbMATHGoogle Scholar
  3. Bai DS, Kim MS, Lee SH (1989) Optimum simple step-stress accelerated life test with censoring. IEEE Trans Reliab 38:528–532CrossRefzbMATHGoogle Scholar
  4. Balakrishnan N (2009) A synthesis of exact inferential results for exponential step-stress models and associated optimal accelerated life-tests. Metrika 69:351–396MathSciNetCrossRefzbMATHGoogle Scholar
  5. Balakrishnan N, Han D (2009) Optimal step-stress testing for progressively Type-I censored data from exponential distribution. J Stat Plan Inference 139:1782–1798MathSciNetCrossRefzbMATHGoogle Scholar
  6. Balakrishnan N, Kundu D (2013) Hybrid censoring: models, inferential results and applications. Comput Stat Data Anal 57:166–209 (with discussions) MathSciNetCrossRefzbMATHGoogle Scholar
  7. Balakrishnan N, Xie Q (2007) Exact inference for a simple step-stress model with Type-I hybrid censored data from the exponential distribution. J Stat Plan Inference 137:3268–3290MathSciNetCrossRefzbMATHGoogle Scholar
  8. Balakrishnan N, Zhang L, Xie Q (2009) Inference for a simple step-stress model with Type-I censoring and lognormally distributed lifetimes. Commun Stat Theory Methods 38:1690–1709MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chung SW, Bai DS (1998) Optimal designs of simple step-stress accelerated life tests for lognormal lifetime distributions. Int J Reliab Qual Saf Eng 5:315–336CrossRefGoogle Scholar
  10. Chung SW, Seo YS, Yun WY (2006) Acceptance sampling plans based on failure-censored step-stress accelerated tests for Weibull distributions. J Qual Maint Eng 12:373–396CrossRefGoogle Scholar
  11. Corana A, Marchesi M, Martini C, Ridella S (1987) Minimizing multi-modal functions of continuous variables with the “simulated annealing” algorithm. ACM Trans Math Softw 13:262–280CrossRefzbMATHGoogle Scholar
  12. Dharmadhikari AD, Rahman MM (2003) A model for step-stress accelerated life testing. Naval Res Logist 50:841–868MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ding C, Yang C, Tse S-K (2010) Accelerated life test sampling plans for the Weibull distribution under Type I progressive interval censoring with random removals. J Stat Comput Simul 80:903–914MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fard N, Li C (2009) Optimal simple step stress accelerated life test design for reliability prediction. J Stat Plan Inference 139:1799–1808MathSciNetCrossRefzbMATHGoogle Scholar
  15. Gouno E, Sen A, Balakrishnan N (2004) Optimal step-stress test under progressive Type-I censoring. IEEE Trans Reliab 53:383–393CrossRefGoogle Scholar
  16. Han D, Balakrishnan N, Sen A, Gouno E (2006) Corrections on “Optimal step-stress test under progressive Type-I censoring”. IEEE Trans Reliab 55:613–614CrossRefGoogle Scholar
  17. Jones MC, Noufaily A (2015) Log-location-scale-log-concave distributions for survival and reliability analysis. Electron J Stat 9:2732–2750MathSciNetCrossRefzbMATHGoogle Scholar
  18. Kateri M, Balakrishnan N (2008) Inference for a simple step-stress model with Type-II censoring, and Weibull distributed lifetimes. IEEE Trans Reliab 57:616–626CrossRefGoogle Scholar
  19. Khamis IH (1997) Optimum \(m\)-step, step-stress design with \(k\) stress variables. Commun Stat Simul Comput 26:1301–1313CrossRefzbMATHGoogle Scholar
  20. Khamis IH, Higgins JJ (1996) Optimum 3-step step-stress tests. IEEE Trans Reliab 45:341–345CrossRefGoogle Scholar
  21. Kiefer J (1974) General equivalence theory for optimum designs (approximate theory). Ann Stat 2:849–879MathSciNetCrossRefzbMATHGoogle Scholar
  22. Lin CT, Chou CC, Balakrishnan N (2013) Planning step-stress test plans under Type-I censoring for the log-location-scale case. J Stat Comput Simul 83:1852–1867MathSciNetCrossRefGoogle Scholar
  23. Lin CT, Chou CC, Balakrishnan N (2014) Planning step-stress test under Type-I censoring for the exponential case. J Stat Comput Simul 84:819–832MathSciNetCrossRefGoogle Scholar
  24. Lin CT, Huang YL, Balakrishnan N (2011) Exact Bayesian variable sampling plans for the exponential distribution with progressive hybrid censoring. J Stat Comput Simul 81:873–882MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ling L, Xu W, Li M (2011) Optimal bivariate step-stress accelerated life test for Type-I hybrid censored data. J Stat Comput Simul 81:1175–1186MathSciNetCrossRefzbMATHGoogle Scholar
  26. Ma H, Meeker WQ (2008) Optimum step-stress accelerated life test plans for log-location-scale distributions. Naval Res Logist 55:551–562MathSciNetCrossRefzbMATHGoogle Scholar
  27. Miller R, Nelson WB (1983) Optimum simple step-stress plans for accelerated life testing. IEEE Trans Reliab 32:59–65CrossRefzbMATHGoogle Scholar
  28. Nelson WB (1990) Accelerated testing: statistical models, test plans, and data analysis. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  29. Nelson WB (2005a) A bibliography of accelerated test plans, part I-overview. IEEE Trans Reliab 54:194–197CrossRefGoogle Scholar
  30. Nelson WB (2005b) A bibliography of accelerated test plans, part II-references. IEEE Trans Reliab 54:370–373CrossRefGoogle Scholar
  31. Nelson WB, Kielpinski TJ (1976) Theory for optimum censored accelerated life tests for normal and lognormal life distributions. Technometrics 18:105–114CrossRefzbMATHGoogle Scholar
  32. Srivastava PW, Shukla R (2008) A log-logistic step-stress model. IEEE Trans Reliab 57:431–434CrossRefGoogle Scholar
  33. Tang LC, Sun YS, Goh TN, Ong HL (1996) Analysis of step-stress accelerated-life-test data: a new approach. IEEE Trans Reliab 45:69–74CrossRefGoogle Scholar
  34. Wu SJ, Lin YP, Chen YJ (2006) Planning step-stress life test with progressively Type-I group-censored exponential data. Stat Neerl 60:46–56MathSciNetCrossRefzbMATHGoogle Scholar
  35. Wu SJ, Lin YP, Chen ST (2008) Optimal step-stress test under Type-I progressive group-censoring with random removals. J Stat Plan Inference 138:817–826MathSciNetCrossRefzbMATHGoogle Scholar
  36. Yeo KP, Tang LC (1999) Planning step-stress life-test with a target acceleration-factor. IEEE Trans Reliab 54:370–373Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsTamkang UniversityNew Taipei CityTaiwan
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

Personalised recommendations