本論文使用伽瑪過程來配適高可靠度產品的衰退過程，並使用一個雙變數的加速退化試驗來加速高可靠度產品的品質衰退速度。假設加速變數與伽瑪過程的形狀參數間的關係服從廣義的Erying模式，我們分別使用 Birnbaum-Saunders分配和逆高斯分配來建立最大概似估計過程。文中使用蒙地卡羅模擬來研究加速退化試驗過程中樣本分配的敏感度。此外，針對本論文建議的加速退化試驗過程，我們建立了關於樣本分配及測量次數的最佳策略，此一最佳策略可極小化受測之高可靠度產品的平均失效時間的漸進變異數，並使得總成本不會超過給定的預算。文中也提出了一個演算法，以利最佳策略之達成。文末使用一個發光二極體的光衰資料為例子，說明所有提出的統計方法的執行過程。

In this thesis, the Gamma process is used to describe the degradation of a highly reliable product, which is subject to a two-variable accelerated degradation test. The relationship between the stress variables and the shape parameter of Gamma process is assumed to follow a generalized Eyring model. Maximum-likelihood estimation process is established based on approximation methods using Birnbaum-Saunders distribution and Inverse Gaussian distribution, respectively. Sensitivity of sample allocation for degradation tests on the maximum-likelihood estimation process is studied using Monte Carlo simulations. Optimal strategy on the sample allocation and measurement frequency for the proposed accelerated degradation test method is developed to minimize the asymptotical variance of the mean time to failure of highly reliable products such that the total cost does not exceed a specified budget. An algorithm is provided to reach the proposed optimal strategy. Finally, a lumen degradation data set of light emitting diodes is presented to illustrate the proposed method.

Contents

1 Introduction...1
1.1 Literature Review...1
1.2 Motivation and Organization...5
1.3 Tables of Acronyms and Abbreviations and Notations...9

2 Interval Estimation for the Mean Time to Failure of Life-
times...13
2.1 The Maximum-Likelihood Estimation...14
2.1.1 Approximation Based on the Birnbaum-Saunders distribution...16
2.1.2 Approximation Based on the Inverse Gaussian distribution...25

3 Planning Constant-Stress Accelerated Degradation Tests...33

4 Sensitivity Analysis and Example...39
4.1 Sensitivity Analysis...39
4.2 Example...41

5 Conclusion...44

6 Appendix A...52

List of Tables
4.1 Bias and mean squared errors of gamma_0_hat, gamma_1_hat, gamma_2_hat, gamma_3_hat and beta_hat...40
4.2 Optimal strategies for the sample size allocation and termination time...43

[1] Doksum, K.-A., H oyland, A. (1992). Models for variable-stress accelerated life testing experiments based on Wiener processes and the inverse Gaussian distribution, Technometrics, 34(1), 74-82.
[2] Durham, S.D., Padgett, W. J. (1997). Cumulative damage models for system failure with application to carbon  bers and composites, Technometrics, 39(1), 34-44.
[3] Folks, J.L., Chhikara, R. S. (1978). The Inverse Gaussian distribution and its statistical application-A review, Journal of the Royal Statistical Society. Series B (Methodological), 40(3), 263-289.
[4] Hogg, R.V., McKean, J. W., Craig, A.T. (2013). Introduction to mathematical statistics, Pearson: Boston.
[5] Komori, Y. (2006). Properties of the Weibull cumulative exposure model, Journal of Applied Statistics, 33(1), 17-34.
[6] Liao, C.-M., Tseng, S.T. (2006). Optimal design for step-stress accelerated degradation tests, IEEE Transactions on Reliability, 55(1), 59-66.
[7] Guan, Q., Tang, Y.-C., (2013). Optimal design of accelerated degradation test based on gamma process models, Chinese Journal of Applied Probability and Statistics, 29(2), 213-224.
[8] Liao, H.T., Elsayed, E.A. (2004). Reliability prediction and testing plan based on an accelerated degradation rate model, International Journal of Materials and Product Technology, 21(5), 402-422.
[9] Lim, H., Yum, B.-J. (2011). Optimal design of accelerated degradation tests based on Wiener process models, Journal of Applied Statistics, 38(2), 309-325.
[10] Lu, C.-J., Meeker, W.Q. (1993). Using degradation measures to estimate a time-to-failure distribution, Technometric, 35(2), 161-174.
[11] Miller, R., Nelson, B. (1983). Optimal simple step stress plans for accelerating life testing, IEEE Transaction on Reliability, R-32(1), 59-65.
[12] Onar, A., Padgett, W.J. (2004). A penalized local D-optimality approach to design for accelerated test models, Journal of Statistical Planning and Inference, 119(2), 411-420.
[13] Park, C., Padgett, W. J. (2005). Accelerated degradation models for failure based on geometric Brownian motion and gamma process, Lifetime
Data Analysis, 11, 511-527.
[14] Padgett, W.J., Tomlinson, M.A. (2004). Inference from accelerated degradation and failure data based on Gaussian process models, Lifetime Data Analysis, 10(2), 191-206.
[15] Pan, Z., Balakrishnan, N. (2010). Multiple-steps step-stress accelerated degradation modeling based on Wiener and gamma processes Communications in Statistics-Simulation and Computation, 39(7), 1384-1402.
[16] Pan, Z., Balakrishnan, N. (2011). Reliability modeling of degradation of products with multiple performance characteristics based on gamma processes, Reliability Engineering and System Safety, 96, 949-957.
[17] Pan, Z., Sun, Q. (2014). Optimal design for step-stress accelerated degradation test with multiple performance characteristics based on gamma processes, Communication in Statistics-Simulation and Computation, 43, 298-314.
[18] Park, C., Padgett, W.J. (2005). Accelerated degradation models for failure based on geometric Brownian motion and gamma process, Lifetime Data Analysis, 11(4), 511-527.
[19] Park, C., Padgett, W.J. (2006). Stochastic degradation models with several accelerating variables, IEEE Transactions on Reliability, 55(2), 379-390.
[20] Park, C., Padgett, W.J. (2007). Cumulative damage models for failure with several accelerating variables, Quality Technology & Quantative Management, 4(1), 17-34.
[21] Park, C., Padgett, W.J. (2008). Cumulative damage models based on gamma processes, In F. Ruggeri, F. Faltin, and R. Kenett, editors, Encyclopedia of Statistics in Quality and Reliability, 1-5, John Wiley & Sons: New York.
[22] Peng, C.-Y., Tseng, S.-T., Tsai, C.-C. (2010). Progressive-stress accelerated degradation test for highly-reliable products, IEEE Transactions on Reliability, 59(1), 30-37.
[23] Shi, Y., Meeker, W.Q. (2012). Bayesian methods for accelerated destructive degradation test planning, IEEE Transactions on Reliability, 61(1), 245-253.
[24] Tsai, T.-R., Lin, C.-W., Sung, Y.-L., Chou, P.-T., Chen, C.-L. and Lio,Y. L. (2012). Inference From Lumen Degradation Data Under Wiener Di usion Process, IEEE Transaction on Reliability, 61(3), 710-718.
[25] Tsai, T.-R., Lio, Y. L., Jiang, N. (2014). Optimal decision on the accelerated degradation test plan under the Wiener process, Quality Technology and Quality Management. To appear.
[26] Tsai, C.-C., Tseng, S.-T., Balakrishnan, N. (2011). Optimal burn-in policy for high reliable products using gamma degradation process, IEEE Transaction on Reliability, 60(1), 234-245.
[27] Tsai, C.-C., Tseng, S.-T., Balakrishnan, N. (2012). Optimal design for degradation tests based on gamma processes with random e ects, IEEE Transactions on Reliability, 61(2), 604-613.
[28] Tsai, C.-C., Tseng, S.-T., Balakrishnan, N. and Lin, C.-T. (2013). Optimal design for accelerated destructive degradation tests, Quality Technology and Quantitative Management, 10(3), 263-276.
[29] Tseng, S.-T., Wen Z.C. (2000). Step-stress accelerated degradation analysis for highly reliable products, Journal of Quality Technology, 32(3), 209-216.
[30] Tseng, S.-T., Peng, C.-Y. (2007). Stochastic di ffusion modeling of degradation data, Journal of Data Science, 5, 315-313.
[31] Tseng, S.-T., Balakrishnan, N., Tsai, C.-C. (2009). Optimal step-stress accelerated degradation test plan for gamma degradation process, IEEE Transactions on Reliability, 58(4), 611-618.
[32] Tseng, S.-T., Tsai, C.-C., Balakrishnan, N. (2011). Optimal sample size allocation for accelerated degradation test based on Wiener process, In: N. Balakrishnan (Editor), Methods and applications of statistics in engineering, quality control, and the physical sciences, Wiley, New York,
330-343.