逆高斯與伽瑪的加速衰變模型之誤判分析

Misspecification Analysis of Inverse Gaussian and Gamma Accelerated Degradation Models

鄧育欣(Yu-Xin Deng)

指導教授 ： 曾勝滄

國立清華大學/理學院/統計學研究所/碩士(2018年)

摘要

加速衰變模型 (accelerated degradation test, ADT) 目前已廣泛地被用來評估高可靠度產品的壽命資訊。當產品存在與壽命具高度相關之品質特徵值 (quality characteristics, QC) 時，則可透過此品質特徵值的衰變量來估計產品壽命資訊 (如產品壽命的 p-quantile)。 由於伽瑪與逆高斯模型之衰變路徑皆為單調遞增，且路徑較為相似，所以時常發生模型的誤判問題，故本研究在加速衰變試驗下探討此兩種模型發生誤判之機率值與對產品壽命 p-quantile估計值所造成的誤判效應。在大樣本下，當伽瑪 (逆高斯) 衰變模型被誤判成逆高斯 (伽瑪) 衰變模型時，本研究先推導出產品壽命 p-quantile的準最大概似估計值 (quasi-maximum likelihood estimate, QMLE) 及其漸近分佈，進而探討它對相對偏誤 (relative bias, RB) 及相對變異 (relative variability, RV) 之誤判效應。其次，在小樣本下，本研究以Device-B資料修改後的模型為實例，分別計算配適此二種衰變模型下的期望均方差 (expected mean square error, EMSE)。具體結論是在絕大多數的樣本數與觀測次數組合中，採用逆高斯加速衰變模型皆有較低的EMSE表現，亦即在此實例下，選用逆高斯加速衰變模型為最佳策略。除此之外，在小樣本的模擬分析，可發現模型誤判時 p-quantile估計值之樣本平均數和樣本變異數，與大樣本下所推得的理論結果極為近似。最後，當樣本數固定且控制模型誤判機率值與 RB 的可容許範圍下，本研究透過使RV極小化，以模擬方法來決定低及高應力觀測次數之配置。結果發現，低(高)應力觀測次數越多(少)，是一較為穩健的ADT設計。

關鍵字

伽瑪加速衰變模型 ； 逆高斯加速衰變模型 ； 誤判機率值 ； 誤判效應 ； 相對偏誤 ； 相對變異 ； 期望均方差

並列摘要

Accelerated degradation tests (ADTs) are widely used to assess the lifetime information (e.g., the p-quantile of the lifetime distribution) of highly reliable products when their quality characteristics (QCs) are closely related to reliability. In practical applications, gamma and inverse Gaussian degradation models are well-known to describe the degradation paths which reveal monotone increasing patterns. Due to possessing very similar degradation paths, these two models are often mis-specified with each other. Therefore, their misspecification analysis shall be an interesting research topic. In this study, we apply White (1982) to address their misspecification effects on the p-quantile of the lifetime distribution. Specifically, in the large sample, we first derive the asymptotic distribution of quasi-maximum likelihood estimation (QMLE) of the p-quantile when the true model is a gamma (inverse Gaussian) accelerated degradation model, but was mis-specified as an inverse Gaussian (gamma) accelerated degradation model. Their misspecification effects on relative bias (RB) and relative variability (RV) are also addressed analytically. Next, we use a modified Device-B dataset to illustrate the misspecification effects between these two degradation models. The results demonstrate that the RBs are not serious for both dual misspecification problems. However, their RVs are not negligible. We also conduct a simulation study to address the performances of misspecification risks. The results demonstrate that adopting inverse Gaussian model has a better performance on the expected mean square error (EMSE). In addition, the RB and RV values for the case of small sample are very similar to that of the case of large sample. Finally, we also propose a simple way to determine the settings of measurement times for both low and high stresses. The results demonstrates that the low-stress (high-stress) level with more (less) measurement times is a good strategy for implementing an ADT plan.

並列關鍵字

gamma accelerated degradation model ； inverse Gaussian accelerated degradation model ； misspecification probability ； misspecification effect ； relative bias ； relative variability ； expected mean square error

參考文獻

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