本文考慮伽瑪隨機過程之階段應力加速衰退試驗(SSADT)之貝氏可靠度分析。在加速因子為溫度之Arrhenius模型下,以主觀先驗分佈經由馬可夫鏈蒙地卡羅方法(MCMC) 得在常溫下產品壽命及可靠度之貝氏推論。另一方面,藉由在類似產品置於正常環境應力水準下之序列衰退試驗中,更新先驗分佈之超參數,以預測產品失效時間之分佈,同時決定測試時間,並以模擬資料驗
證所提方法的可行性和準確性以及貝氏方法之穩健性。

Degradation analysis is more efficient than the conventional life tests in drawing reliability assessment for high quality products. This thesis aims on the Bayesian approach to the degradation test when the degradation data of different products are collected under higher than normal stress levels via independent gamma processes. Reliability inference of the population under normal condition will be made based on the posterior distribution of the underlying parameters with the aid of Markov chain Monte Carlo method. Further sequentially predictive inference on individual reliability under normal condition is also proposed. Simulation study is presented to show the
appropriateness of the proposed methods, and the robustness of the prior distribution.

參考文獻
[1] Abramowitz, M. and Stegun, I. (1972). Handbook of Mathematical Functions: with Formulas,
Graphs, and Mathematical Tables, Dover, New York.
[2] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis (Second Edition),
Springer-Verlag, New York.
[3] Birnbaum, Z. W. and Sunders, S. C. (1969). A new family of life distributions. Journal of
Applied Probability, 6, 319–327.
[4] Doksum, K. A. and Normand, S.-L. T. (1995). Gaussian models for degradation process-part
I: Methods for the analysis of biomarker data. Lifetime Data Analysis, 1, 131–144.
[5] Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple se-
quences. Statistical Science, 7, 457–511.
[6] Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian
restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6,
721–741.
[7] Graves, T. L., Hamada, M. S., Klamann, R., Koehler, A., and Martz, H. F. (2007). A fully
Bayesian approach for combining multi-level informaion in multi-state fault tree quantification.
Reliability Engineering and System Safety, 92, 1476–1483.
[8] Hamada, M., Martz, H. F., Reese, C. S., Graves, T., Johnson, V., and Wilson, A. G. (2004). A
fully Bayesian approach for combining multilevel failure information in fault tree quantification
and optimal follow-on resource allocation. Reliability Engineering and System Safety, 86, 297–
305.
[9] Hasting, W. K. (1970). Monte Carlo sampling methods using Markov chains and their appli-
cations. Biometrika, 57, 97–109.
[10] Li, M. and Meeker, W. Q. (2014). Application of Bayesian methods in reliability data analyses.
Journal of Quality Technology, 46, 1–23.
[11] Liao, C. M. and Tseng, S. T. (2006). Optimal design for step-stress accelerated degradation
tests. IEEE Transactions on Reliability, 55, 59–66.
[12] Lim, H. and Yum, B. J. (2011). Optimal design of accelerated degradation tests based on
Wiener process models. Journal of Applied Statistics, 38, 309–325.
[13] Meeker, W. Q. and Escobar, L. A. (1993). A review of recent research current issues in
accelerated testing. International Statistical Review, 61, 147–168.
[14] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data, Wiley, New
York.
[15] Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis,
John Wiley & Sons, New York.
[16] Padgeet, W. and Tomlinson, M. A. (2004). Inference from accelerated degradation and failure
data based on Gaussian process models. Lifetime Data Analysis, 10, 191–206.
[17] Park, C. and Padgett, W. J. (2005). Accelerated degradation models for failure based on
geometric Brownian motion and gamma processes. Lifetime Data Analysis, 11, 511–527.
[18] Pettit, L. I. and Young, K. D. S. (1999). Bayesian analysis for inverse Gaussian lifetime data
with measures of degradation. Journal of Statistical Computation and Simulation, 63, 217–
234.
[19] Ross, S. M. (2006). Simulation (Fourth Edition), Elsevier Academic Press, U.S.A.
[20] Touw, A. E. (2009). Bayesian estimation of mixed Weibull distributions. Reliability Engineering
and System Safety, 94, 463–473.
[21] Tseng, S. T., Balakrishnan, N., and Tsai, C. C. (2009). Optimal step-stress accelerated degra-
dation test plan for gamma degradation processes. IEEE Transactions on Reliability, 58,
611–618.
[22] Tseng, S. T. and Wen, Z. C. (2000). Step-stress accelerated degradation analysis for highly
reliable products. Journal of Quality Technology, 32, 209–216.
[23] Wang, L., Pan, R., Li, X., and Jiang, T. (2013). A Bayesian reliability evaluation method
with integrated accelerated degradation testing and field information. Reliability Engineering
and System Safety, 112, 38–47.
[24] Whitmore, G. A. (1995). Estimating degradation by a Wiener diffusion process subject to
measurement error. Lifetime Data Analysis, 1, 307–319.
[25] Whitmore, G. A., Crowder, M. J., and Lawless, L. F. (1998). Failure inference from a marker
process based on a bivariate Wiener model. Lifetime Data Analysis, 4, 229–251.
[26] Whitmore, G. A. and Schenkelberg, F. (1997). Modeling accelerated degradation data using
Wiener diffusion with a scale transformation. Lifetime Data Analysis, 3, 27–45.