針對高可靠產品的壽命推估問題，加速試驗（Accelerated Test）是業界常用的研究方法。加速試驗因資料的收集型態不同，可區分為加速壽命試驗 (Accelerated Life Test) 及加速衰變試驗 (Accelerated Degradation Test)。一般而言，執行加速試驗的成本非常昂貴，因此如何在成本限制下，有效率地決定實驗配置，方可獲得產品壽命的精確估計值是產品製造商的重要決策問題。
本研究針對加速試驗之最適配置問題進行研究，內容包含以下三個部分：
(i) 首先我們提出指數分散 (exponential dispersion, ED) 衰變模型來描述高可靠度產品的品質特徵值之衰變路徑，其優點不僅涵蓋 Wiener、Gamma 或Inverse Gaussian 等常用的連續型衰變模型，亦可描述離散型之衰變路徑如compound Poisson 模型。欲使產品壽命的第q 百分位數估計值之近似變異數極小化，在兩個與三個應力的加速衰變試驗中，針對ED加速衰變模型，探討加速衰變試驗之樣本數配置及應力水準的最適設計問題。
(ii) 當加速衰變模型為指數分散加速衰變模型時，在實驗之總成本限制下，本研究更進一步提出一演算法來決定加速衰變試驗的總樣本數、量測次數及總量測時間等之最適決策問題，方可獲得最精確的產品壽命估計值。
(iii) 文獻上有關加速試驗之最適配置問題，大多假設模型中之參數為已知的情況下進行研究；然而，模型中的真正參數值在實際應用中大多為未知值。藉由實驗者的實務經驗或現有的實驗資料，本研究利用貝氏分析 (Bayesian analysis) 之方法，提出一依序設計（sequential design）之策略以執行有效率之加速壽命試驗。本研究並將此方法應用至具有輕質量、高強度、高耐度等優點的高分子複合材料（polymer composite materials）之加速疲勞試驗（accelerated fatigue tests），以有效率地收集材料之失效資料進而分析並預測其在實際應用上的表現。

Accelerated tests are widely used to assess the lifetime information for highly reliable products. Due to different types of collected data, the accelerated tests can be further classified as accelerated life tests (ALTs) and accelerated degradation tests (ADTs). For highly reliable products, conducting an accelerated test is very costly. To obtain the precise prediction of lifetime information, how to design an efficient planning under cost constraints is a critical task. For the planning of accelerated tests, this study includes the following three topics:
(i) Several researchers have attempted to address the problem of determining the sample size allocation and the settings of stress levels, but their results have been based only on specific degradation models. Therefore, they lack a unified approach toward general degradation models. We first proposes a class of exponential dispersion (ED) degradation models which include some special cases such as the Wiener, gamma, and inverse Gaussian processes. Assuming that the underlying degradation path comes from the ED class, we analytically derive the optimum allocation rules by minimizing the asymptotic variance of the estimated q quantile of product’s lifetime for two-level and three-level ADT allocation problems no matter the testing stress levels are prefixed.
(ii) Assuming that the underlying degradation path comes from the ED class, we further determine the total sample size, the number of measurements within a degradation path, and the total testing times simultaneously under the constraints of total experimental cost. For this constrained optimization, we propose an algorithm to determine the optimum design by minimizing the asymptotic variance of the q quantile of the product’s lifetime.
(iii) There has been a lot of development in optimum test planning, most of the methods assume that the true parameter values are known. However, in reality, the true model parameters may depart from the planning values. Therefore, we use Bayesian framework and propose a sequential test planning strategy for ALTs. Furthermore, we apply the proposed strategy to the accelerated cyclic fatigue tests of polymer composite materials which are lightweight and comparable levels of strength and endurance. We also use extensive simulation to study the properties of the proposed sequential test planning strategy.

1 Introduction . . . . . . . . . . . . . . . . 1
1.1 Background .................................... 1
1.2 Motivation..................................... 2
1.3 The Layout of the Dissertation ......................... 3
2 Literature Review . . . . . . . . . . . . . . . . 6
2.1 Planning ALTs and Related Literature ..................... 6
2.2 Planning ADTs and Related Literature..................... 7
2.2.1 Degradation Models ........................... 7
2.2.2 Planning Accelerated Degradation Tests . . . . . . . . . . . . . . . . 8
3 Optimum Allocation Rule for Accelerated Degradation Tests with a Class
of Exponential Dispersion Degradation Models . . . . . . . . . . . . . . . . 10
3.1 A Class of Exponential Dispersion Degradation Models . . . . . . . . . . . . 10
3.2 Assumptions and Problem Formulation..................... 11
3.3 Optimum ED-ADT Allocation Rule....................... 13
3.3.1 Expression of Asymptotic Variance of ξˆ . . . . . . . . . . . . . . . . 13
3.3.2 Optimum ED-ADT Allocation Rule When All Stresses Are Prefixed . 15
3.3.3 Optimum ED-ADT Allocation Rule for k = 2 When One Stress Level
Is Prefixed................................. 18
3.4 Two Illustrative Examples ............................ 19
3.4.1 Example1: Stress Relaxation Data (Yang, 2007) . . . . . . . . . . . 19
3.4.2 Example2: DeviceBData(Meeker & Escobar, 1998) . . . . . . . . . 23
3.4.3 Optimum Two-Level Allocation Rule When One Stress Level Is Free to Be Optimized ............................. 25
3.5 Efficiency Gain Compared to Non-optimum Allocation Rules . . . . . . . . . 28
3.5.1 Comparison With Two Well-Known Plans Under k = 2 . . . . . . . . 28
3.5.2 Comparison With Some Well-Known Compromise Plans for k= 3 . . 29
3.6 Conclusion..................................... 30
4 Planning Accelerated Degradation Tests under the Constraint of Total Cost 32 4.1 Problem Formulation............................... 32
4.2 First Passage Time of an ED-ADT Model ................... 34
4.3 Planning Methodology .............................. 38
4.3.1 The Exact Expression of Asymptotic Variance . . . . . . . . . . . . . 38
4.3.2 The Constrained Optimum Plan..................... 40
4.4 An Algorithm for Finding the Optimum Test Plan. . . . . . . . . . . . . . . 41
4.5 Illustrative Example ............................... 44
4.5.1 Stress Relaxation Data Revisited .................... 44
4.5.2 Test Plans under Different Degradation Models . . . . . . . . . . . . 45
4.5.3 Relative Efficiencies for the Case of Equal Frequency under All Stress
Levels ................................... 48
4.6 Sensitivity Analysis................................ 50
4.7 Conclusion..................................... 53
5 Sequential Test Planning for Polymer Composites . . . . . . . . . . . . . . . . 54
5.1 Introduction.................................... 54
5.1.1 Background ................................ 54
5.1.2 Motivation................................. 56
5.2 Model ....................................... 56
5.2.1 Physical and Statistical Models ..................... 56
5.2.2 Maximum Likelihood Estimation and Asymptotic Variance . . . . . . 58
5.3 Test Planning Methodology ........................... 59
5.3.1 Prior Distribution............................. 60
5.3.2 Criterion for Sequential Bayesian Designs. . . . . . . . . . . . . . . . 60
5.3.3 Design Optimization ........................... 61
5.4 Application on Polymer Composites Test Planning . . . . . . . . . . . . . . 64
5.4.1 Parameter Estimation .......................... 64
5.4.2 Sequential Bayesian Design with Limited Number of Historical Data . 67
5.5 Design Performance Evaluation ......................... 69
5.5.1 Simulation Procedure........................... 70
5.5.2 Simulation Results ............................ 71
5.5.3 Comparison on Different Historical Data . . . . . . . . . . . . . . . . 74
5.6 Conclusion..................................... 75
6 Conclusion and the Topics for Future Research . . . . . . . . . . . . . . . . 77
Appendices . . . . . . . . . . . . . . . . 79
Appendix A . . . . . . . . . . . . . . . . 80
A.1 The proof of Equation (3.7) ........................... 80
A.2 The proof of Result2............................... 81
A.3 The proof of Result3............................... 84
A.4 The proof of Result4............................... 84
Appendix B . . . . . . . . . . . . . . . . 87
B.1 Proof of Step 5 in Section 4.4 .......................... 87
B.2 Sensitivity Analysis on Estimated Values of λ . . . . . . . . . . . . . . . . . 88
Appendix C . . . . . . . . . . . . . . . . 91
C.1 Formulation of Fisher information matrix.................... 91
C.2 Performance of MCMC.............................. 94
Bibliography.............................. 95

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