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姓名	林祐頡(Yu-Chieh Lin)  查詢紙本館藏	畢業系所	統計研究所
論文名稱	穩定過程之衰變分析 (Stable Processes for Degradation Analysis)
相關論文	★ 加速不變原則之偏斜-t過程 ★ Hougaard 過程之衰變分析
檔案	[Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   至系統瀏覽論文 (2025-7-29以後開放)
摘要(中)	對於高可靠度產品的壽命推論, 衰變分析已成為建立統計模型之最有效和最重要的技術。隨機過程則廣泛應用於衰變資料之分析, 然而對於具有厚尾特徵的衰變路徑, 文獻上則是付之闕如。本文提出涵蓋常見的維納與柯西過程為特例的非線性穩定過程, 配適具厚尾性質之衰變資料, 建立具有隨機效應之(加速) 衰變模型, 以評估產品壽命分配之相關資訊。關於首次通過門檻值之時間分配(即壽命分配) 的估計, 則以兩近似公式為猜想, 並與以模擬為驗證基礎的產品壽命分配做比較。佐以三組實例分析呈現產品壽命的百分位數估計、相對應的逐點拔靴法信賴區間及模型的適合度檢定等。
摘要(英)	Degradation analysis has become the most effective and important technique for establishing statistical models to draw lifetime inference of highly reliable products. Stochastic processes are widely used to analyze degradation tests data. However, there is relatively few literature on degradation paths with heavy-tailed characteristics. This thesis proposes a non-linear stable process, which not only has heavy-tail but also covers the common Wiener and Cauchy processes as special cases, to establish degradation model with possibly random effects to assess the information of product’s lifetime distribution. Two conjecturable formulas are used to approximate the distribution function of the first passage time (product’s lifetime) which are also compared with the estimated life time distribution based on Monte-Carlo simulation. The proposed model is applied to three real datasets in which the estimation of the quantiles of the products lifetime distribution along with the corresponding pointwise bootstrap confidence intervals and the goodnessof-fit tests are demonstrated.
關鍵字(中)	★ Birnbaum-Saunders分配 ★ 首次通過門檻值之時間點 ★ 非奇異第二型渥爾 特拉積分方程式 ★ 厚尾特徵 ★ 學生-t 過程	關鍵字(英)	★ Birnbaum-Saunders distribution ★ first passage time ★ heavy-tail ★ non-singular second-kind Volterra integral equation ★ Student’s t process
論文目次	目錄 摘要 i Abstract ii 誌謝 iii 目錄 iv 圖目錄 vi 表目錄 viii 第一章緒論 1 1.1 背景與研究動機. . . . . . . . . . . . . . . . . . . 1 1.2 文獻回顧. . . . . . . . . . . . . . . . . . . . . . 2 1.3 研究方法. . . . . . . . . . . . . . . . . . . . . . 5 1.4 本文架構. . . . . . . . . . . . . . . . . . . . . . 6 2 第二章非線性穩定過程之統計推論 7 2.1 穩定分配. . . . . . . . . . . . . . . . . . . . . . 7 2.2 穩定過程. . . . . . . . . . . . . . . . . . . . . 11 2.3 非線性穩定過程衰變模型之參數估計. . . . . . . . . . . 12 2.4 衰變模型之適合度檢定. . . . . . . . . . . . . . . . 13 2.5 拔靴法信賴區間. . . . . . . . . . . . . . . . . . . 14 3 第三章首次通過門檻值之時間分配與猜測 15 3.1 首次通過門檻值之時間分配. . . . . . . . . . . . . . . 15 3.2 首次通過門檻值之時間分配近似函數. . . . . . . . . . . 18 3.3 首次通過門檻值之時間分配比較. . . . . . . . . . . . . 21 4 第四章實例分析 23 4.1 火車車輪資料分析. . . . . . . . . . . . . . . . . . 25 4.2 合金材料疲勞試驗之資料分析. . . . . . . . . . . . . . 31 4.3 紅外發光二極管資料分析. . . . . . . . . . . . . . . 36 第五章結論與未來研究 43 參考文獻 44 圖目錄 1.1 火車車輪資料。. . . . . . . . . . . . . . . . . . . 3 1.2 車輪資料獨立增量之常態、柯西、學生-t 及穩定分配分位-分位圖與 Anderson-Darling (AD) 檢定結果。. . . . . . . . . . . 3 4.1 火車車輪資料之虛擬失效時間、壽命之累積分配函數估計及其95% 逐點拔靴信賴區間；其中水平實線與曲線交點對應的時間即為t0.05 之點估計與區間估計。. . . . . . . . . . . . . . . . . . . 29 4.2 火車車輪資料中ˆ FT (t) 與˜ FT (t) 比較。. . . . . . 30 4.3 金屬合金疲勞裂縫資料之衰變路徑圖。. . . . . . . . . . 33 4.4 金屬合金疲勞裂縫資料之虛擬失效時間、壽命之累積分配函數及其95% 逐點拔靴法信賴區間；其中水平實線與曲線交點對應的時間即 t0.05 之點估計與區間估計。. . . . . . . . . . . . . . . 34 4.5 金屬合金疲勞裂縫資料中ˆ FT (t) 與˜ FT (t) 比較。. . . 35 4.6 金屬合金疲勞裂縫資料中ˆ FT (t) 與ˇ FT (t) 比較。. . . 35 4.7 紅外發光二極管資料之衰變路徑圖。. . . . . . . . . . . 37 4.8 紅外發光二極管資料之虛擬失效時間、壽命之累積分配函數及其95% 逐點拔靴法信賴區間。. . . . . . . . . . . . . . . . . . 40 4.9 320 毫安培下紅外發光二極管之壽命分配函數估計結果。. . . 41 4.10 170 毫安培下紅外發光二極管之壽命分配函數估計結果。. . 42 4.11 50 毫安培下紅外發光二極管之壽命分配函數估計結果。. . . 42 表目錄 3.1 給定(δ, ω) = (5, 1) 之下, ˆ FT (t)、˜ FT (t) 和ˇ FT (t) 之t0.1 估計結果。. . . . . . . . . . . . . . . . . 21 3.2 給定(δ, ω) = (5, 1) 之下, ˜ FT (t) − ˆ FT (t) 和ˇ FT (t) − ˆ FT (t) 之t0.1 估計結果。 22 3.3 ˆ FT (t)、˜ FT (t) 與ˇ FT (t) 之優、缺點整理。. . . 22 4.1 火車車輪資料配適維納過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . 28 4.2 火車車輪資料配適伽瑪過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . 28 4.3 火車車輪資料配適逆高斯過程之參數估計、L(ˆΘ) 、AIC 與BIC 值。. . 28 4.4 火車車輪資料配適學生-t 過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . 28 4.5 火車車輪資料配適正態穩定過程之參數估計、L(ˆ Θ)、AIC 與BIC 值。. . 28 4.6 火車車輪資料配適柯西過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . 29 4.7 火車車輪資料配適穩定過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . 29 4.8 火車車輪資料中ˆ FT (t)( ˜ FT (t)) 之百分位數估計與95% 逐點拔靴信賴區間(ω = 770) 。. . . . . . . . . . . . . . . 30 4.9 金屬合金疲勞裂縫資料配適維納過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . . .32 4.10 金屬合金疲勞裂縫資料配適伽瑪過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . . .33 4.11 金屬合金疲勞裂縫資料配適逆高斯過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . .33 4.12 金屬合金疲勞裂縫資料配適學生-t 過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . .33 4.13 金屬合金疲勞裂縫資料配適正態穩定過程之參數估計、L(ˆΘ)、AIC 與 BIC 值。. . . . . . . . . . . . . . . . . . . . 34 4.14 金屬合金疲勞裂縫資料配適柯西過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . . .34 4.15 金屬合金疲勞裂縫資料配適穩定過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . . .34 4.16 金屬合金疲勞裂縫資料中ˆ FT (t)、˜ FT (t) 與ˇ FT (t) 之百分位數估計與95%逐點拔靴信賴區間(ω = 70)。. . . . . . . .36 4.17 紅外發光二極管資料配適維納過程之參數估計、L(ˆ Θ)、AIC 與BIC 值。 39 4.18 紅外發光二極管資料配適學生-t 過程之參數估計、L(ˆΘ)、AIC 與BIC 值。. . . . . . . . . . . . . . . . . . . . . . .39 4.19 紅外發光二極管資料配適穩定過程之參數估計、L(ˆ Θ)、AIC 與BIC 值。 39 4.20 紅外發光二極管資料之三種適合度檢定p 值(ω = 3000)。. . 39 4.21 紅外發光二極管資料中各環境應力下ˆ FT (t) 之百分位數估計與95% 逐點拔靴信賴區間(ω = 3000)。. . . . . . . . . . . . .41
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指導教授	樊采虹 彭健育(Tsai-Hung Fan Chien-Yu Peng)	審核日期	2020-8-11
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