針對高可靠度產品的壽命推估問題，傳統上大多採用溫度、濕度或電壓做為加速變數。惟上述方法只適用於固態產品，最近 Yao, Tseng and Wong (2017) 提出以酸鹼值 (pH value) 作為加速變數來推估液態產品奈米溶膠 (nano-sol) 的保存期限 (shelf-life)。在這篇文章中有關粒徑分布之建模工作是透過混合常態分布 (mixture normal distribution) 來執行。一般而言，混合Log-F分布更適合對粒徑分布進行建模。因此，本研究建構出混合Log-F分布之pH值加速衰變模型，透過期望條件最大演算法 (expectation/conditional maximization algorithm) 求得模型中未知參數的最大概似估計值，進而預測產品於正常使用條件下之保存期限及其95% 信賴區間。具體而言，以pH 8.5及8.75的衰變試驗資料來預測pH 8.9下之保存期限的驗證實驗中，我們可發現粒徑分布採用右偏態的雙Log-F混合分布，在預測的表現上 (點估計及信賴區間) 均顯著優於 Yao, et al. (2017)。除此之外，當粒徑分布真實來自於混合Log-F分布，卻誤判為混合常態分布時，對產品保存期限估計值之準確度 (accuracy) 與精確度 (precision) 的影響頗為巨大，其相對偏誤及相對變異分別高達11.06% 及15.53。

Conventionally, temperature, humidity or voltage accelerations are widely used for assessing the lifetime information of highly-reliability products. The approaches are only applicable to the solid products. Recently, by using a pH acceleration model, Yao, Tseng and Wong (2017) introduced a statistical model for predicting the shelf-life of nano-sol (liquid-type product). In their model, a mixture normal distribution is adopted for modeling the particle size distribution. Generally speaking, a mixture Log-F distribution is more appropriate to model particle size distribution of each group. Therefore, this thesis deals with the inference problem of a pH accelerated model where the particle size distribution follows a mixture Log-F distribution. Maximum likelihood estimator (MLE) and expectation/conditional maximization (ECM) algorithm are applied to predict the shelf-life of nano-sol (under normal use condition) and its corresponding 95% confidence interval. In the validation experiment, by using the testing data of pH 8.5 and pH 8.75 to predict the shelf-life at pH 8.9, it demonstrated that the prediction performance (point estimation and confidence interval) of our proposed procedure is better than that of Yao, Tseng and Wong (2017). Furthermore, we also use an analytical approach to address the influence of accuracy and precision on the shelf-life prediction when the particle size distribution is wrongly mis-specified from a mixture Log-F distribution to a mixture normal distribution. From the real case study, we found that the relative bias (RB) and relative variation (RV) may be up to 11.06% and 15.53, respectively. It means that the model mis-specification of particle size distribution is a critical issue and cannot be negligible.

摘要.........i
Abstract.........ii
致謝辭.........iii
目錄.........iv
圖目錄.........vi
表目錄.........vii
第一章 緒論.........1
1.1 前言.........1
1.2 研究主題與動機.........2
1.3 研究架構.........2
第二章 文獻回顧與問題描述.........3
2.1 動機例子.........3
2.2 保存期限定義與pH值加速概念.........3
2.3 混合常態分布之pH值加速衰變模型.........6
2.4 Log-F分布介紹.........9
2.5 模型誤判理論.........10
第三章 混合Log-F分布之pH值加速衰變模型建構.........13
3.1 問題描述.........13
3.2 pH值加速衰變模型架構.........14
3.2.1 (k,r) 之選擇.........14
3.2.2 p(t | S), μ_1 (t | S), σ_1 (t | S), μ_2 (t | S) 及 σ_2 (t | S) 之展開式.........17
3.3 pH值加速衰變模型之期望條件最大演算法.........22
3.4 壽命估計及其信賴區間.........24
3.5 動機例子回顧.........24
3.6 模擬與驗證.........26
3.6.1 模擬實驗.........26
3.6.2 驗證實驗.........27
第四章 混合Log-F分布之pH值加速衰變模型誤判分析.........28
4.1 問題描述.........28
4.2 模型誤判分析.........29
4.3 動機例子回顧.........30
第五章 總結與後續研究.........34
附錄.........35
附錄A I(θ) 之計算.........35
附錄B v(θ) 之計算.........41
附錄C v_N 之計算.........43
附錄D C(θ_F,θ_N^*) 之計算.........45
參考文獻.........50

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