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系統識別號 U0002-1707201311472500
中文論文名稱 偏常態量測誤差模型下之加速破壞衰變試驗
英文論文名稱 Accelerated Destructive Degradation Test based on Skew-Normal Measurement Error Model
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 101
學期 2
出版年 102
研究生中文姓名 林姿吟
研究生英文姓名 Tzu-Yin Lin
學號 600190259
學位類別 碩士
語文別 中文
口試日期 2013-07-04
論文頁數 46頁
口試委員 指導教授-蔡志群
委員-林千代
委員-彭健育
中文關鍵字 加速破壞衰變試驗  偏常態分配  最佳試驗配置 
英文關鍵字 accelerated destructive degradation test  skew-normal distribution  optimal test plan 
學科別分類 學科別自然科學數學
中文摘要 加速破壞衰變試驗,其量測過程中,需破壞測試樣本,以便量測到產品的品質特徵值,並配合提高環境應力,以加速產品衰變,進而有效地推估產品的壽命資訊。Tsai et al. (2013) 建構一量測誤差服從常態分配的非線性 ADDT 衰變模型。然而,當量測誤差為非對稱的分配時,此時使用偏常態分配來描述較為合適。因此,本文以聚合物材料為動機例子,首先建構一偏常態 ADDT 衰變模型,並探討其壽命資訊,其結果顯示使用本文所建構的衰變模型,來推估產品壽命較為精確。接下來,模擬一組偏常態 ADDT 衰變資料,並執行最佳化設計,其結果顯示在不同預算下,偏常態 ADDT 衰變模型的近似變異數皆較小。最後,模擬分析結果可知,模擬結果與理論結果是相近的,而模型誤判對於產品壽命的推估影響很大。
英文摘要 Accelerated destructive degradation tests (ADDTs) that measurement process of quality characteristic (QC) would destroy the tested units at higher stress environment are powerful and useful tools for lifetime assessment of highly reliable products. Motivated by a polymer data, Tsai et al. (2013) proposed a nonlinear ADDT model with measurement error that follows a normal distribution. However, the skew normal distribution that generalizes the normal distribution to allow for non-zero skewness is more appropriate for describing the measurement error with non-symmetrical pattern. Hence, this article used skew-normal distribution to construct the nonlinear ADDT model. The results show that the skew normal ADDT model has better precision on the estimated lifetime of the products than normal ADDT model. Moreover, the optimal design problem based on the proposed ADDT model is discussed on this article. Finally, Monte Carlo simulations are used to evaluate the asymptotical results and the effect of model misspecification on the estimator of the products’ lifetime.




論文目次 1 緒論 1
1.1 前言 . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 文獻探討 . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 非破壞衰變模型 . . . . . . . . . . . 3
1.2.2 破壞衰變模型 . . . . . . . . . . . . . 4
1.2.3 偏常態分配 . . . . . . . . . . . . . . . 5
1.3 研究動機與目的 . . . . . . . . . . . . . . . 8
1.4 研究架構 . . . . . . . . . . . . . . . . . . . . . 13
2 問題描述 15
3 最佳化試驗配置 18
3.1 AVar(t_p)之推導 . . . . . . . . . . . . . . . 18
3.2 成本函數 . . . . . . . . . . . . . . . . . . . . . 19
3.3 最佳化模型 . . . . . . . . . . . . . . . . . . . 20
3.4 最佳試驗配置求解步驟 . . . . . . . . . . 21
4 ADDT 資料分析 23
4.1 實例資料分析 . . . . . . . . . . . . . . . . . 23
4.2 最佳化設計 . . . . . . . . . . . . . . . . . . . 26
4.3 模擬與模型誤判分析 . . . . . . . . . . . . 29
5 結論及後續研究 36
附錄一 38
附錄二 42
參考文獻 43
參考文獻 [1] Azzalini, A. (1985). “A class of distributions which includes the normal ones,” Scandinavian Journal of Statistics, vol. 12, 171-178.
[2] Boulanger, M. and Escobar, L. A. (1994). “Experimental design for a class of accelerated degradation tests,” Technometrics, vol. 36, 260-272.
[3] Carey, M. B. and Koenig, R. H. (1991). “Reliability assessment based on accelerated degradation,” IEEE Transactions on Reliability, vol. 40, 499-506.
[4] Escobar, L. A., Meeker, W. Q., Kugler, D. L., and Kramer, L. L. (2003). “Accelerated destructive degradation tests: data,models, and analysis,” Chapter 21 in Mathematical and Statistical Methods in Reliavility, Lindqvist, B. H. and Doksum, K. A., Editors, River Edge, NJ: World Scientific Publishing Company.
[5] Gomez, H. W. and Salinas, H. S. (2010). “Information matrix for generalized skew-normal distributions,” Proyecciones Journal of Mathematics, vol. 29, 83-92.
[6] Gupta, R. C. and Brown, N. (2001). “Reliabilitystudies of the skew-normal distribution and its application to a strengthstress model,” Communications in Statistics - Theory and Methods, vol. 30, 2427-2445.
[7] Henze, N. (1986). “A probabilistic representation of the skew-normal distribution,” Scandinavian Journal of Statistics, vol. 13, 271-275.
[8] Jafari, H. and Hashemi, R. (2011). “Optimal designs in a simple linear regression with skew-normal distribution for error term,” Applied Mathematics, vol. 1, 65-68.
[9] Jeng, S. L., Huang, B. Y., and Meeker, W. Q. (2011). “Accelerated destructive degradation tests robust to distribution misspecification,” IEEE Transactions on Reliability, vol. 60, 701-711.
[10] Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. New York: John Wiley & Sons.
[11] Monti, A. C. (2003). “A note on the estimation of the skew normal and the skew exponential power distributions,” Metron - International Journal of Statistics, LXI, 205-219.
[12] Nelson, W. (1981). “Analysis of performance degradation data from accelerated tests,” IEEE Transactions on Reliability, vol. 30, 149-155.
[13] Nelson, W. (1990). Accelerated Testing: Statistical Models, Test Plans, and Data Analysis. New York: John Wiley & Sons.
[14] Peng, C. Y. and Tseng, S. T. (2009). “Mis-specification analysis of linear degradation models,” IEEE Transactions on Reliability, vol. 58, 444-455.
[15] Peng, C. Y. and Tseng, S. T. (2013). “Statistical lifetime inference with skew-wiener linear degradationmodels,” IEEE Transactions on Reliability, vol. 62, 338-350.
[16] Shi, Y. and Meeker, W. Q. (2012). “Bayesian methods for accelerated destructive degradation test planning,” IEEE Transactions on Reliability, vol. 61, 245-253.
[17] Shi, Y., Meeker, W. Q., and Escobar, L. A. (2009). “Accelerated destructive degradation test planning,” Technometrics, vol. 51, 1-13.
[18] Tsai, C. C., Tseng, S. T., Balakrishnan, N., and Lin, C. T. (2013). “Optimal design for accelerated destructive degradation test,” Quality Technology and Quantitative Management, vol. 10, 263-276.
[19] Yang, G. (2007). Life Cycle Reliability Engineering. Hoboken, New Jersey: John Wiley & Sons.
[20] Yu, H. F. (2003). “Designing an accelerated degradation experiment by optimizing the estimation of the percentile,” Quality and Reliability Engineering International, vol. 19, 197-214.
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