Optimal Step Stress Accelerated Life Testing for the Length-Biased Exponential Cumulative Exposure Model

Abstract

This paper considers a simple step stress accelerated life test for units modeled by a length-biased exponential distribution. The cumulative exposure model of time to failure holds in this accelerated life test model. The optimal test plan is constructed by determining the optimal stress change time. Parameters of the model are estimated by using the maximum likelihood estimation method. The corresponding approximate confidence intervals are obtained by using the asymptotic normality features of the maximum likelihood estimators. Theoretical outcomes are illustrated with simulation studies and a real data example.

Keywords

• Accelerated life test
• Cumulative exposure model
• Length-biased exponential distribution
• Maximum likelihood estimation
• Optimal test plan
• Step-stress model

References

1. Abbas, K. and Firdos, M. (2018). Optimum step stress accelerated life testing for Frechet distribution. Journal of Reliability and Statistical Studies, 11(1), 29-44.
2. Adamidis K. and Loukas S. (1998). A lifetime distribution with decreasing failure rate. Statistics & Probability Letters, 39:35-42.
3. Akhter, Z., Saran, J., Verma, K. and Pushkarna, N. (2020). Moments of Order Statistics from Length-Biased Exponential Distribution and Associated Inference. Annals of Data Science, 1-26.
4. Chung, S.W. and Bai, D.S. (1998). Optimal designs of simple step-stress accelerated life tests for lognormal lifetime distributions, International Journal of Reliability, Quality and Safety Engineering, 5(04), 315-336.
5. Dara S.T. and Ahmad M. (2012). Recent advances in moment distribution and their hazard rates. Academic Publishing GmbH KG, Lap Lambert.
6. Ebrahem, M.A.H. and Al-Masri, A. Q. (2007). Optimum simple step-stress plan for log-logistic cumulative exposure model, Metron-International Journal of Statistics, 65(1), 23-34.
7. Godambe, V. P. (1960). An optimum property of regular maximum likelihood estimation. The Annals of Mathematical Statistics, 31(4), 1208-1211.
8. Gradshteyn, I. S., and Ryzhik, I. M. (2014). Table of integrals, series, and products. Academic press.

9. Gupta R.D. and Kundu D. (1999). Theory & Methods: Generalized exponential distributions. The Australian & New Zealand Journal of Statistics 41(2):173-188.
10. Haghighi, F. (2014). Optimal design of accelerated life tests for an extension of the exponential distribution. Reliability Engineering & System Safety, 131, 251-256.
11. Ma, H. and Meeker, W. Q. (2008). Optimum step-stress accelerated life test plans for log-location-scale distributions. Naval Research Logistics (NRL), 55(6), 551-562.
12. Miller, R. and Nelson, W. (1983). Optimum simple step-stress plans for accelerated life testing. IEEE Transactions on Reliability, 32(1), 59-65.
13. Nelson, W.B. (1980). Accelerated life testing-step-stress models and data analyses. IEEE transactions on reliability, 29(2), 103-108.
14. Nelson, W.B. (1990). Accelerated testing: statistical models, test plans, and data analysis (Vol. 344). John Wiley & Sons.
15. Team, R. C. (2013). R: A language and environment for statistical computing.
16. Saxena, S., Zarrin, S., Kamal, M. and Ul-Islam, A. (2012). Optimum Step Stress Accelerated Life Testing for Rayleigh Distribution. International journal of statistics and applications, 2(6), 120-125.
17. Tarvirdizade, B., and Ahmadpour, M. (2016). Estimation of the stress-strength reliability for the two-parameter bathtub-shaped lifetime distribution based on upper record values. Statistical Methodology, 31,58-72.