臺灣博碩士論文加值系統

在一般的壽命測試和衰變測試中，若在實驗期間無法看到足夠的產品失效，則無法對於產品壽命提供足夠的資訊，此時便會進而考慮重複量測的加速衰變測試。在非線性的退化模型假設下，本文處理此種測試規劃的最佳化問題，同時，測試單元間變異亦以非線性之隨機項存在於模型中。

由於模型的非線性型態，概似函數與費雪訊息矩陣，以及退化模型所衍生出的壽命分佈之推算有其難處。因此，本文以蒙地卡羅積分方法計算測試規劃最佳化之準則，即參數估計量的漸進變異數，並以格子點搜索的方式找到最佳的測試規劃。為了減輕計算上的負擔，本文將最佳化問題考慮在一些限制條件下進行處理，同時並使用擬蒙地卡羅法及一些R套件來進行有效率的統計計算。本文亦將所提出之方法應用在一集成電路裝置之實例上。

Repeated measures accelerated degradation tests (RMADTs) can provide more information about the product reliability
when one would expect few or even no failures during a study. In this paper, we deal with the optimal test planning problem under the non-linear Arrhenius acceleration model. Further, unit-to-unit variability is described by the random effects, which is also non-linear in time.

Due to the assumption of non-linear mixed effect model (NLMEM), the analytical calculation of the likelihood and some related functions such as Fisher information matrix are difficult. Therefore, we perform the Monte Carlo integration to calculate the asymptotic variance of estimators as the optimality criterion. Grid search procedures are conducted to find the optimum test plan under some constraints. Quasi-random low-discrepancy sequences and some R packages for efficient computation are used to relief the computational burden. The method is illustrated with an application of an integrated circuit device example.

1 Introduction 1
1.1 Background and Motivation . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Planning the RMADTs . . . . . . . . . . . . . . . . . 2
1.2.2 Evaluation of Fisher Information Matrix under NLMEMs 3
1.3 Motivating Example and Previous Work . . . . . . . . . . . . 3
1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Degradation Models and Lifetime Distribution 6
2.1 Accelerated Degradation Model . . . . . . . . . . . . . . . . 6
2.1.1 General Degradation Path Model . . . . . . . . . . . . 6
2.1.2 Arrhenius Acceleration Factor . . . . . . . . . . . . . 6
2.1.3 Re-parameterization . . . . . . . . . . . . . . . . . . 7
2.2 Likelihood, MLE and Fisher Information . . . . . . . . . . . 9
2.3 Lifetime Distribution for Degradation Models . . . . . . . . . 10
2.3.1 Relationship between Degradation and Failures . . . . 10
2.3.2 p Quantile of Lifetime Distribution . . . . . . . . . . 12
3 Planning RMADTs 13
3.1 Planning Information and Constraints . . . . . . . . . . . . . 13
3.2 Specification of RMADT Plans . . . . . . . . . . . . . . . . . 13
3.2.1 Stress Level and Test Units Allocation . . . . . . . . . 13
3.2.2 Inspection Schedule . . . . . . . . . . . . . . . . . . 14
3.3 Criterion for Choosing a Plan . . . . . . . . . . . . . . . . . . 15
3.3.1 Variance Covariance Matrix of Parameters Estimators 15
3.3.2 Asymptotic Variance of Estimator of p Quantile . . . . 16
4 Methodology in Computation 18
4.1 Monte Carlo Integration . . . . . . . . . . . . . . . . . . . . 18
4.2 Quasi Monte Carlo Integration . . . . . . . . . . . . . . . . . 18
5 Application 22
5.1 Computational Performance and Efficiency . . . . . . . . . . 22
5.1.1 Evaluation of Likelihood and Information . . . . . . . 24
5.1.2 Evaluation of Lifetime Distribution and Derivatives
of F . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Grid Search for Optimal RMADT Plans . . . . . . . . . . . . 33
5.2.1 Re-allocation of Device-B Example . . . . . . . . . . 33
5.2.2 Optimal Choice of Number of Stress Levels . . . . . . 34
5.3 Validation of Optimality with Finite Sample . . . . . . . . . . 39
5.3.1 Variability of Estimators . . . . . . . . . . . . . . . . 39
5.3.2 Convergence of Variance with Increasing Sample Size 39
6 Conclusions and Future Work 42
Bibliography 44
Appendix 47
Appendix A Derivation of Fisher Information 47
Appendix B Derivation of AVar of tp 50
Appendix C Figures and Tables 52

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