Design and Analysis of Gamma Degradation Process

• Main Authors: Tsai, Chih-Chun, 蔡志群
• Other Authors: Tseng, Sheng-Tsaing
• Format: Others
• Language: zh-TW
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/61361459034356092864

博士 === 國立清華大學 === 統計學研究所 === 97 === Nowadays, degradation analysis has been widely used to assess the lifetime information of highly reliable products. In the literature, degradation models are mostly built by the randomeffect models and/orWiener process. However, for materials that lead to fatigue data, it is more appropriately modeled by a gamma process which exhibits a monotonic increasing pattern. Hence, it is of great interest to design and analyze the degradation tests based on gamma process. In this thesis, we study the following three topics. (1) Motivated by laser data,we discuss the effects ofmodelmis-specificationwhen the truemodel comes fromgamma degradationprocess, and iswrongly treated as Wiener degradation process. We derive the asymptotic distribution of quasi MLE (QMLE) so that the penalty formodelmis-specification can be addressed sequentially. The result demonstrates that the effect on the accuracy of the product’smean-time-to failure (MTTF) prediction strongly depends on the ratio of critical value and scale parameter of gamma degradation process. Besides, the effects on the precision of the product’s MTTF prediction are critical when the shape and scale parameters of gamma degradation process are large enough. Furthermore,we use a simulationstudy to address the penalty of model mis-specification. The simulation result is quite close to the theoretical result even when the sample size and termination time are not large enough. (2) Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. Firstly, we propose the SSADT model based on a gamma process with a linear degradation path. Under the constraint that the total experimental cost not exceeding a pre-specified budget, the optimal test plan can be obtained by minimizing the approximate variance of the estimated MTTF of the lifetime distribution of the product. The sensitivity analysis reveals that the optimal test plan is quite robust to moderate departures from the model parameters. Furthermore,we address the effect ofmodelmis-specification under SSADT. The results show that the effect on the accuracy of product’s MTTF prediction is not so critical, while the effect on the precision of product’s MTTF prediction is substantial when gamma degradation process ismis-specified as Wiener degradation process. (3) Burn-in test has been widely used to eliminate latent failures or weak components in the factory before the products are shipped to the customers. The traditional burn-in test over a short period of time to collect time-to-failure data is rather inefficient. This decision problem can be solved by degradation models. We first propose amixture gamma process to describe the degradation path of the product. Next, a decision rule for classifying a unit as a normal or a weak unit is proposed. Finally, a cost model is used to determine the optimal burn-in time. Furthermore,we address themis-specification issue of treating a mixture gamma degradation process as amixtureWiener degradation process by a simulation study. The results reveal that the effect on the probabilities of misclassification is significant.