Bayesian accelerated life testing models for the log-normal and gamma distributions under dual-stresses
Keywords:
Accelerated life testing, Bayes, Generalized Eyring relationship, Markov chain Monte Carlo, Reliability
Abstract
In this paper, a Bayesian approach to accelerated life testing models with two stressors is presented. Lifetimes are assumed to follow either a log-normal distribution or a gamma distribution, which have been mostly overlooked in the Bayesian literature when considering multiple stressors. The generalized Eyring relationship is used as the time transformation function, which allows for the use of one thermal stressor and one non-thermal stressor. Due to the mathematically intractable posteriors of these models, Markov chain Monte Carlo methods are utilized to obtain posterior samples on which to base inference. The models are applied to a real dataset, where model comparison metrics are calculated and estimates are provided of the model parameters, predictive reliability, and mean time to failure. The robustness of the models is also investigated in terms of the prior specification.References
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Soyer, R., Erkanli, A., & Merrick, J. R. 2008. Accelerated life tests: Bayesian models. Encyclopedia of Statistics in Quality and Reliability, 1, 20–30.
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van der Linde, A. 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society Series B: Statistical Methodology, 64(4), 583–639.
Sun, T., & Shi, Y. 2016. Estimation for Birnbaum-Saunders distribution in simple step stress-accelerated life test with type-II censoring. Communications in Statistics - Simulation and Computation, 45, 880– 901.
Taketomi, N., Yamamoto, K., Chesneau, C., & Emura, T. 2022. Parametric distributions for survival and reliability analyses, a review and historical sketch. Mathematics, 10(20), 3907. https://doi.org/10.3390/math10203907.
Upadhyay, S. K., & Mukherjee, B. 2010. Bayes analysis and comparison of accelerated Weibull and accelerated Birnbaum-Saunders models. Communications in Statistics - Theory and Methods, 39, 195–213.
Brooks, S. P., & Gelman, A. 1998. General methods for monitoring convergence of iterative simulations. Journal of Computational and Graphical Statistics, 7(4), 434–455.
Burnham, K. P., & Anderson, D. R. 1998. Model Selection and Inference. New York: Springer.
Chaloner, K., & Larntz, K. 1992. Bayesian design for accelerated life testing. Journal of Statistical Planning and Inference, 33, 245–259.
Escobar, L. A., & Meeker, W. Q. 2006. A review of accelerated test models. Statistical Science, 21(4), 552–577.
Fujita, M., Jomjunyong, S., Itoh, Y., Yokoyama, S., & Matsumoto, S. 1997. Predicting service life of a bright Ni-Cr electroplating system on steel subjected to the CASS test. Transactions of the IMF, 75(3), 98–100.
Gilks, W. R., & Wild, P. 1992. Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41(2), 337–348.
Gilks, W. R., Best, N. G., & Tan, K. K. C. 1995. Adaptive rejection Metropolis sampling within Gibbs sampling. Applied Statistics, 44(4), 455–472.
Gupta, A., Ranjan, R., & Upadhyay, S. K. 2023. A Bayes analysis and comparison of Arrhenius Weibull and Arrhenius lognormal models under competing risk. American Journal of Mathematical and Management Sciences, 42(2), 105–125.
Karim, M. R., & Suzuki, K. 2007. Analysis of warranty data with covariates. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 221(4), 249–255.
Kececioglu, D. B. 2002. Reliability and Life Testing Handbook. Lancaster: DEStech Publications.
Mazzuchi, T. A., Soyer, R., & Vopatek, A. L. 1997. Linear Bayesian inference for accelerated Weibull model. Lifetime Data Analysis, 3, 63–75.
Moustafa, K., Hu, Z., Mourelatos, Z. P., Baseski, I., & Majcher, M. 2021. System reliability analysis using component-level and system-level accelerated life testing. Reliability Engineering and System Safety, 214, 107755. https://doi.org/10.1016/j.ress.2021.107755.
Mukhopadhyay, C., & Roy, S. 2016. Bayesian accelerated life testing under competing log-location-scale family of causes of failure. Computational Statistics, 31, 89–119.
Neal, R. M. 2003. Slice sampling. The Annals of Statistics, 31(3), 705–767.
Nelson, W. B. 1990. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. New York: Wiley.
Nelson, W. B., & Kielpinski, T. J. 1976. Theory for optimum censored accelerated life tests for normal and lognormal life distributions. Technometrics, 18(1), 105–144.
Owen, W. J., & Padgett, W. J. 2000. A Birnbaum-Saunders accelerated life model. IEEE Transactions on Reliability, 49(2), 224–229.
Polson, N. G., & Soyer, R. 2017. Augmented probability simulation for accelerated life test design. Applied Stochastic Models in Business and Industry, 33, 322–332.
Smit, N., & Raubenheimer, L. 2022a. Bayes factors for accelerated life testing models. Communications for Statistical Applications and Methods, 29(5), 513–532.
Smit, N., & Raubenheimer, L. 2022b. Bayesian accelerated life testing: A generalized Eyring-Birnbaum-Saunders model. Quality and Reliability Engineering International, 38(1), 195–210.
Smit, N., Raubenheimer, L., Mazzuchi, T., & Soyer, R. 2024. A Bayesian generalized Eyring-Weibull accelerated life testing model. Quality and Reliability Engineering International, 40(2), 1110–1125.
Soyer, R., Erkanli, A., & Merrick, J. R. 2008. Accelerated life tests: Bayesian models. Encyclopedia of Statistics in Quality and Reliability, 1, 20–30.
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van der Linde, A. 2002. Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society Series B: Statistical Methodology, 64(4), 583–639.
Sun, T., & Shi, Y. 2016. Estimation for Birnbaum-Saunders distribution in simple step stress-accelerated life test with type-II censoring. Communications in Statistics - Simulation and Computation, 45, 880– 901.
Taketomi, N., Yamamoto, K., Chesneau, C., & Emura, T. 2022. Parametric distributions for survival and reliability analyses, a review and historical sketch. Mathematics, 10(20), 3907. https://doi.org/10.3390/math10203907.
Upadhyay, S. K., & Mukherjee, B. 2010. Bayes analysis and comparison of accelerated Weibull and accelerated Birnbaum-Saunders models. Communications in Statistics - Theory and Methods, 39, 195–213.
Published
2025-03-21
How to Cite
Smit, N. (2025). Bayesian accelerated life testing models for the log-normal and gamma distributions under dual-stresses. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2293
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Section
Research Articles
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