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中文論文名稱 Burr XII分配之型I設限資料的逐步部分加速測試:MLE在小樣本的估計效率
英文論文名稱 Step Partially Accelerated Life Tests for Burr XII Distribution with Type I Censored Data:MLE Performance in Small Sample
校院名稱 淡江大學
系所名稱(中) 統計學系碩士班
系所名稱(英) Department of Statistics
學年度 100
學期 2
出版年 101
研究生中文姓名 劉家成
研究生英文姓名 Chia-Cheng Liu
電子信箱 king90409@hotmail.com
學號 699650320
學位類別 碩士
語文別 中文
口試日期 2012-05-23
論文頁數 36頁
口試委員 指導教授-陳景祥
委員-歐士田
委員-吳錦全
中文關鍵字 部分加速壽命測試  加速因子  Burr XII分配  最大概似估計  型I設限資料 
英文關鍵字 Partially accelerated life test  Acceleration factor  Burr XII distribution  Maximum Likelihood Estimation  Type I censored data 
學科別分類 學科別自然科學統計
中文摘要 隨著產品可靠性的提升, 在一般條件下的壽命試驗會花費很多時間, 進而耗掉很多成本, 唯一的辦法就是降低樣本個數, 此外要短時間內得到產品的資料也相當困難。為了確保實驗單位能加速失效進而縮短實驗時間, 加速壽命測試經常被用來做為進行壽命實驗的主要架構, 但在一些狀況下,失效模型的加速因子參數是未知狀況下, 特別是當試驗對象為新產品時, 部分加速壽命測試則是另一個很好的選擇。
在小樣本下, 本文研究考慮逐步應力部分加速壽命測試方法, 並選擇以Burr XII 分配做為壽命分配, 使用最大概似估計法估計分配及加速因子的參數。分配及加速因子的參數估計值皆是經由推導的聯立方程式計算求得, 除了參數估計外, 本文也同時推導近似變異數且計算參數的信賴區間以比較估計量的實際表現。
英文摘要 The lifetime of products under normal conditions endures a long period of time because of high reliability of products. Reducing the sample size is the only way to solve this problem. It is difficult to obtain information about the lifetime of products under normal conditions. In order to assure rapid failure and then shorten the testing period, accelerated life test (ALT) is commonly used to obtain information about the lifetime of products quickly. In some cases, the failure model relating the lifetime of products and the stress are not known or cannot be assumed, especially for the new products. So step-stress partially accelerated life tests are the optional choice.
This article considers step-stress partially accelerated life tests for small samples under Burr XII distribution. We use maximum likelihood method to estimate the distribution parameters and acceleration factor. The estimates are obtained by solving two nonlinear equations simultaneously. In addition, we also perform simulation of the asymptotic variance and confidence intervals of the parameters to evaluate the empirical performances of our estimators.
論文目次 目錄
第一章緒論.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1
1.1 研究背景.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1
1.2 研究動機與目的.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 1
1.3 本文架構.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 3
第二章文獻探討及相關研究.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4
2.1 加速壽命測試.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 4
2.2 逐步部分加速壽命測試.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 6
2.3 Optimization 估計方法.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 8
第三章理論推導.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9
3.1 壽命模型之介紹.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 9
3.2 測試模型.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 10
3.3 最大概似估計法.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 12
3.4 漸近變異與共變異.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 16
第四章模擬研究.... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 19
4.1 模擬步驟.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 20
4.2 結果討論.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 21
第五章結論及建議.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 23
參考文獻.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25
圖目錄
2.1 三種不同應力加速方法. . . . . . . . . . . . . . . . . . . . . . 5
2.2 Optimization 五種估計法. . . . . . . . . . . . . . . . . . . . . 8
3.3 逐步部分加速測試示意圖. . . . . . . . . . . . . . . . . . . . . 11
表目錄
4.1 方法比較(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 方法比較(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3 參數估計-第一組參數. . . . . . . . . . . . . . . . . . . . . . . 31
4.4 參數估計-第二組參數. . . . . . . . . . . . . . . . . . . . . . . 32
4.5 估計漸近變異與共變異(第一組參數) . . . . . . . . . . . . . . . . 33
4.6 估計漸近變異與共變異(第二組參數) . . . . . . . . . . . . . . . . 34
4.7 近似信賴區間(第一組參數) . . . . . . . . . . . . . . . . . . . . . 35
4.8 近似信賴區間(第二組參數) . . . . . . . . . . . . . . . . . . . . . 36
參考文獻 [1] Attia, A.F., Abel-Ghaly, A.A. and Abel-Ghani, M.M.(1996). The Estimation Problem of Partially Accelerated Life Tests for Weibull distribution by Maximum likelihood method with Censored Data. Proceeding of the 31st Annual Conference of Statistics, Computer Sciences and Operation
Research, ISSR, Cairo University, 128-138.
[2] Abdel-Ghaly, A.A., Attia, A.F. and Abdel-Ghani, M.M. (1997). The Bayesian Estimation of Weibull Parameters in Step Partially Accelerated life Tests with Censored Data. Proceeding of the 32st Annual Conference of Statistics , Computer Sciences and Operation Rresearch, ISSR, Cairo
University, 45-59.
[3] Abdel-Ghaly, A.A., Attia, A.F. and Abdel-Ghani, M.M. (2002).The Maximum Likelihood Estimates in Step Partially Accelerated life Tests for the Weibull Parameters in Censored Data. Commincation in Statistics-Theory
and Methods, 31(4): 551-573.
[4] Abdel-Ghaly, A.A., El-Khodary, E.H. and Ismail, A.A. (2002). Maximum Likelihood Estimation and Optimal Design in Step-Stress Partially Accelerated life Tests for the Pareto distribution with Type I Censoring.Proceeding of the 14st Annual Conference on Statistics and Computer Modeling in Human and Social Sciences, Faculty of Economics and Political Science, Cairo University, 16-29.
[5] Abdel-Ghaly, A.A., El-Khodary, E.H. and Ismail, A.A. (2003). Estimation and Optimal Design in Step Partially Accelerated life Tests for the Compound Pareto distribution using Type II Censoring. Proceeding of the 15st Annual Conference on Statistics and Computer Modeling in Human
and Social Sciences, Faculty of Economics and Political Science, Cairo University, 35-47.
[6] Abdel-Ghani, M.M. (2004). The Estimation Problem of the Log Logistic Parameters in Step Partially Accelerated Life Tests using Type I Censored Data The National Review of Social Sciences, 41(2): 1-19.
[7] Abd-Elfattah, A.M., Soliman, A.H. and Nassr, S.G. (2008) . Estimation in Step-Stress Partially Accelerated Life Tests for the Burr Type XII distribution Using Type I censoring. Statistically Methodology,5(6): 502-514.
[8] Abdel-Ghaly, A.A., Attia, A.F. and Abdel-Ghani, M.M. (1996). The Maximum Likelihood Estimates in Step Partially Accelerated Life Tests for the Weibull Parameters in Censored Data. Communication in Statistics -Theory and Methods, 31(4), 551-573.
[9] Abdullah M. Abd-Elfattah and Abdullah M. Al-Harbey. (2010).Inferences for Burr Parameters Based on Censored Samples in Accelerated Life Tests. Department of Statistics, Faculty of Science King Abdulaziz University,Jeddah, Saudi Arabia.
[10] Bai, D.S., Chung, S.W. and Chun, Y.R.(1993) Optimal design of partially accelerated life tests for the lognormal distribution under Type-I censoring.
[11] Bhattacharyya, G.K. and Soejoeti, Z. (1989). A Tampered Failure Rate Model for Step-Stress Accelerated Life Test. Communication in Statistics Theory and Methods, 8: 1627-1644.
[12] Bai, D.S. and Chung, S.W. (1992). Optimal Design of Partially Accelerated Life Tests for the Exponential distribution under Type I Censoring.IEEE Trans. Reliability , 41: 400-406.
[13] Cohen, A.C.(1965). Maximum likelihood estimation in the Weibull Distribution based on complete and on censored samples. Technometrics, 5 :579-588.
[14] DeGroot, M.H. and Goel, P.K. (1979). Bayesian estimation and optimal Design in Partially Accelerated life Testing. Naval Research Logistics Quarterly, 16(2) 223-235.
[15] Goel, P.K. (1971). Some Estimation problems in the study of Tampered Random Variables. Thesis Department of Statistics, Cranegie-Mellon University,Pittspurgh, Pennsylvania.
[16] Ismail, A.A. (2004). The test design and parameter estimation of Pareto lifetime distribution under partially accelerated life tests. Ph.D. Department of Statistics, Faculty of Economics and Political Science, Cairo University
, Egypt.
[17] Ismail, A.A. (2006).On the Optimal Design of Step-Stress Partially Accelerated Life Tests for the Gompertz Distribution with Type I Censoring. InterStat, Elctronic Journal.
[18] Meeker, William Q. (1998), Statistical methods for reliability data. John Wiley & Sons: New York.
[19] Nelson,W. (1990), Accelerated Life Testing: Statistical Models. Test Plans and Data Analysis. John Wiley & Sons: New York.
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