本論文針對奈米溶膠產品的保存期限進行相關壽命推論工作。有別於傳統在非液態高可靠度產品上，採用溫度、溼度或電壓來執行加速壽命/衰變實驗，本研究以酸鹼 (pH) 值作為加速因子來推估奈米溶膠一類的液態高可靠度產品之保存期限。
奈米溶膠中奈米顆粒之分布為一典型的直方圖值 (histogram-valued) 資料，本論文提出以雙常態混合模型來描述奈米顆粒分布，進而建構出pH值加速衰變模型來完整描述在不同pH值下，奈米溶膠內的奈米顆粒分布隨時間之動態變化情形；然後透過期望條件最大演算法 (Expectation/Conditional Maximization algorithm) 來求得模型中未知參數的最大概似估計值，進而預測產品於正常使用條件下之保存期限及其95%信賴區間。
本論文亦探討在試驗總成本不超過事先給定的預算下，尋找pH值加速衰變試驗之最佳試驗配置 (optimal testing plan)，使產品保存期限估計值之近似變異數可極小化；亦即決定在每個pH值下的最佳量測次數、量測頻率及其樣本數。藉由敏感度分析 (sensitivity analysis) 可發現本研究所提的最佳試驗計畫之表現頗為穩健。
最後，本論文探討了奈米顆粒分布發生誤判時，對於產品保存期限估計值之準確度 (accuracy) 與精確度 (precision) 的影響。具體來說，本研究推導出當模型發生誤判時，產品保存期限之估計量的大樣本漸近分布，藉此結果可探討模型誤判對於產品保存期限之影響。由實證研究可發現當模型發生誤判實，其對於產品保存期限估計值之準確度及精確度的影響皆不容忽視。

Motivated by nano-sol data set, this thesis addresses the shelf-life prediction of a nano-sol product. Instead of using temperature, humidity or voltage to do accelerated life/degradation experiments in the past, this study used pH as accelerating factor to predict the shelf-life of nano-sol product.
The observed distribution data of nano-particles in the nano-sol are histogram-valued frequencies. A mixture of two normal distributions was used to describe the particle size distribution. The time evolution of the particle size distribution under different pH values was described by a pH accelerated degradation model. The maximum likelihood estimator (MLE) for unknown parameters in the pH accelerated degradation model can be solved by applying Expectation/Conditional Maximization (ECM) algorithm. The estimated shelf-life under normal-use-condition and corresponding 95% confidence intervals can then be obtained.
An optimal test plan for pH accelerated degradation model can be obtained by minimizing the approximate variance of the estimated shelf-life of the nano-sol product under the constraint that the total experimental cost not exceeding a pre-specified budget. The sensitivity analysis reveals that the optimal test plan is quite robust to moderate departures from the model parameters.
Finally, the effects of model mis-specification were discussed. Specifically the asymptotic large sample distribution of the shelf-life estimates was derived when the particle size distribution described in the first topic is wrongly fitted with inappropriate model. The result show that the effects on the accuracy and precision of the product’s shelf-life are critical.

[1] 彭健育(2008). “高可靠度產品之衰變試驗分析,”國立清華大學統計學研究所博士論文.
[2] Abdel-Hameed, M. (1975). “A gamma wear process,” IEEE Transactions on Reliability, 24, 152–153.
[3] Bae, S. J., Kim, S. J., Kim, M. S., Lee, B. J. and Kang, C. W. (2008). “Degradation analysis of nano-contamination in plasma display panels,” IEEE Transactions on Reliability, 57, 222–229.
[4] Bae, S. J. and Kvam, P.H. (2004). “A nonlinear random-coefficients model for degradation testing,” Technometrics, 46, 460–469.
[5] Bagdonavicius, V. and Nikulin, M. (2002). Accelerated Life Models: Modeling and Statistical Analysis, Chapman & Hall/CRC, New York.
[6] Balka, J., Desmond, A. F. and McNicholas, P. D. (2009). “Review and implementation of curemodels based on first hitting times forWiener processes,” Lifetime Data Analysis, 15, 147–176.
[7] Basu, A. K., & Wasan, M. T. (1974). “On the first passage time processes of brownian motion with positive drift,” Scandinavian Actuarial Journal, 1974, 144-150.
[8] Berman, M. (1981). “Inhomogeneous and modulated gamma processes,” Biometrika, 68, 143–152.
[9] Billard, L. and Diday, E. (2006). Symbolic Data Analysis: Conceptual Statistics and Data Mining, Wiley & Sons, New York.
[10] Birnbaum, Z. W. and Saunders, S. C. (1969). “A new family of life distributions,” Journal of Applied Probability, 6, 319–327.
[11] Boulanger, M. and Escobar, L. A. (1994). “Experimental design for a class of accelerated degradation tests,” Technometrics, 36, 260–272.
[12] Chao, M. T. (1999). “Degradation analysis and related topics: Some thoughts and a review,” The Proceedings of the National Science Council. A, 23, 555–566.
[13] Chen, Z. and Zheng, S. (2005). “Lifetime distribution based degradation analysis,” IEEE Transactions on Reliability, 54, 3–10.
[14] Chhikara, R. S. and Folks, L. (1989). The Inverse Gaussain Distribution. Theory, Methodology, and Applications, Marcel Dekker, New York.
[15] Crowder, M. and Lawless, J. F. (2007). “On a scheme for predictive maintenance,” European Journal of Operational Research, 16, 1713–1722.
[16] Dobson, A. J. and Barnett, A. (2008). An introduction to generalized linear models. CRC press.
[17] Doksum, K. A. and Hόyland, A. (1992). “Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution,” Technometrics, 34, 74-82.
[18] Doksum, K. A. and Normand, S. L. T. (1995). “Gaussian models for degradation processes - Part I: methods for the analysis of biomarker data,” Lifetime Data Analysis, 1, 135–144.
[19] Horrocks, J. and Thompson, M. E. (2004). “Modeling event times with multiple outcomes using the Wiener process with drift,” Lifetime Data Analysis, 10, 29–49.
[20] Hsieh, M. H. and Jeng, S. L. (2007). “Accelerated discrete degradation models for leakage current of ultra-thin gate oxides,” IEEE Transactions on Reliability, 56, 369–380.
[21] Ishwaran, I. and James, L. F. (2004). “Computational methods for multiplicative intensity models using weighted gamma processes: proportional hazards, marked point processes, and panel count data,” Journal of the American Statistical Association, 99, 175–190.
[22] Kalbfleisch, J. D. and Prentice, R. I. (2002). The Statistical Analysis of Failure Time Data, John Wiley & Sons, New York.
[23] Lawless, J. F. (2002). Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York.
[24] Lawless, J. F. and Crowder, M. (2004). “Covariates and random effects in a gamma process model with application to degradation and failure,” Lifetime Data Analysis, 10, 213–227.
[25] Lee, M. L. T. and Whitmore, G. A. (1993). “Stochastic processes directed by randomized time,” Journal of Applied Probability, 30, 302–314.
[26] Li, Q. and Kececioglu, D. B. (2004). “Optimal design of accelerated degradation tests,” International Journal of Materials and Product Technology, 20, 73–90.
[27] Liao, C. M. and Tseng, S. T. (2006). “Optimal design for step-stress accelerated degradation tests,” IEEE Transactions on Reliability, 55, 59–66.
[28] Lindstrom, M. J. and Bates, D. M. (1990). “Nonlinear mixed effects models for repeated measures data,” Biometrics, 46, 673–687.
[29] Lu, J. C. and Meeker, W. Q. (1993). “Using degradation measures to estimate a time-to-failure distribution,” Technometrics, 35, 161–173.
[30] McLachlan, G. J., & Jones, P. N. (1988). “Fitting mixture models to grouped and truncated data via the EM algorithm,” Biometrics, 44, 571-578.
[31] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data, John Wiley & Sons, New York.
[32] Meeker, W. Q., Escobar, L. A. and Lu, C. J. (1998). “Accelerated degradation tests: modeling and analysis,” Technometrics, 40, 89–99.
[33] Moran, P. A. P. (1954). “A probability theory of dams and storage systems,” Australian Journal of Applied Science, 5, 116–124.
[34] Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, John Wiley & Sons, New York.
[35] Noortwijk, J. M. V. (2009). “A survey of the application of gamma processes in maintenance,” Reliability Engineering & System Safety, 94, 2–21.
[36] Padgett, W. J. and Tomlinson, M. A. (2004). “Inference from accelerated degradation and failure data based on Gaussian process models,” Lifetime Data Analysis, 10, 191–206.
[37] Park, C. and Padgett, W. J. (2005). “Accelerated degradation models for failure based on geometric Brownian motion and gamma process,” Lifetime Data Analysis, 11, 511–527.
[38] Park, C. and Padgett, W. J. (2006). “Stochastic degradation models with several accelerating variables,” IEEE Transactions on Reliability, 55, 379–390.
[39] Park, S. J., Yum, B. J. and Balamurali, S. (2004). “Optimal design of step stress degradation tests in the case of destructive measurement,” Quality Technology & Quantitative Management, 1, 105–124.
[40] Pascual, F. G. (2005). “Maximum likelihood estimation under misspecified lognormal and Weibull distributions,” Communications in Statistics – Simulation and Computation, 34, 503–524.
[41] Pascual, F. G. (2006). “Accelerated life test plans robust to misspecification of the stress-life relationship,” Technometrics, 48, 11–25.
[42] Pascual, F. G. and Montepiedra, G. (2003). “Model-robust test plans with applications in accelerated life testing,” Technometrics, 45, 47–57.
[43] Pascual, F.G. and Montepiedra, G. (2005). “Lognormal and Weibull accelerated life test plans under distribution misspecification,” IEEE Transactions on Reliability, 54, 43–52.
[44] Peng, C. Y. and Tseng, S. T. (2009). “Mis-specification analysis of linear degradation models,” IEEE Transactions on Reliability, 58, 444–455.
[45] Seshadri, V. (1999). Inverse Gaussian Distribution: Statistical Theory and Applications, Springer-Verlag, New York.
[46] Shiau, H. J. J. and Lin, H. H. (1999). “Analyzing accelerated degradation data by nonparametric regression,” IEEE Transactions on Reliability, 48, 149–158.
[47] Singpurwalla, N. D. (1995). “Survival in dynamic environments,” Statistical Science, 10, 86–103.
[48] Tseng, S. T., Hamada, M. S. and Chiao, C. H. (1995). “Using degradation data from a fractional factorial experiment to improve fluorescent lamp reliability,” Journal of Quality Technology, 27, 363–369.
[49] Tseng, S. T. and Liao, C. M. (1998). “Optimal design for a degradation test,” International Journal of Operations and Quantitative Management, 4, 293–301.
[50] Tseng, S. T. and Peng, C. Y. (2007). “Stochastic diffusion modeling of degradation data,” Journal of Data Science, 5, 315–333.
[51] Tseng, S. T. and Yu, H. F. (1997). “A termination rule for degradation experiments,” IEEE Transactions on Reliability, 46, 130–133.
[52] Wasan, M. T. (1968). “On an inverse Gaussian process,” Scandinavian Actuarial Journal, 1968, 69-96.
[53] White, H. (1982). “Maximum likelihood estimation of misspecified models,” Econometrica, 50, 1–25.
[54] Whitmore, G. A. (1995). “Estimating degradation by a Wiener diffusion process subject to measurement error,” Lifetime Data Analysis, 1, 307–319.
[55] Whitmore, G. A. and Schenkelberg, F. (1997). “Modelling accelerated degradation data using Wiener diffusion with a time scale transformation,” Lifetime Data Analysis, 3, 27–45.
[56] Wu, C. J. and Chang, C. T. (2002). “Optimal design of degradation tests in presence of cost constraint,” Reliability Engineering & System Safety, 76, 109–115.
[57] Yu, H. F. (2007). “Mis-specification analysis between normal and extreme value distribution for a linear regression model,” Communications in Statistics – Theory and Methods, 36, 499–521.
[58] Yu, H. F. and Tseng, S. T. (1998). “On-line procedure for determining an appropriate termination time for an accelerated degradation experiment,” Statistica Sinica, 8, 207–220.
[59] Yu, H. F. and Tseng, S. T. (1999). “Designing a degradation experiment,” Naval Research logistics, 46, 689–706.
[60] Yu, H. F. and Tseng, S. T. (2004). “Designing a degradation experiment with a reciprocal Weibull degradation rate,” Quality Technology & Quantitative Management, 1, 47–63.