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中文論文名稱 評估單因子隨機效應模型中多種單邊容許界限在非常態隨機分布下的表現
英文論文名稱 An Evaluation of Various One-Sided Tolerance Limits for One-Way Random Effects Model under Non-Normal Distributions
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 104
學期 2
出版年 105
研究生中文姓名 張凱雲
研究生英文姓名 Kai-Yun Chang
學號 601190258
學位類別 碩士
語文別 中文
口試日期 2016-06-02
論文頁數 55頁
口試委員 指導教授-陳順益
委員-賴耀宗
委員-吳秀芬
中文關鍵字 單因子隨機效應模型  非常態隨機變數  伽瑪分布  單邊容許界限  蒙地卡羅模擬方法 
英文關鍵字 One-Way Random Effects Model  Non-Normal Distributions  Gamma Distribution  One-Sided Tolerance Limits  Monte Carlo Simulation 
學科別分類 學科別自然科學數學
中文摘要 本文評估單因子隨機效應模型中多種容許界限在非常態隨機變數資料的表現。我們使用Mee和Owen(1983)、Vangel(1992)、Krishnamoorthy和Mathew(2004) 、Harris和Chen (2006a) 及Chen和Harris (2006b)這五篇論文所建構的七種容許界限,並利用蒙地卡羅模擬方法求得不同情況下各種容許界限的模擬涵蓋率、平均值與標準差,就模擬方法所產生的結果進行比較與總結。模擬的結果顯示,根據常態分布的性質所建構出的容許界限,應用於伽瑪分布的數據時導致大部份的情況下涵蓋率表現不如預期。如需改善則需要根據在單因子隨機效應模型下推導出適用於伽瑪分布或其它非對稱性機率分布的容許界限。
英文摘要 This paper gives the results from a computer simulation study concerning the estimated coverage rate and the average of the tolerance limits with their standard deviation for seven one-sided tolerance limits under the one-way random effects model when data are generated from non-normal distributions. These procedures are from Mee and Owen (1983), Vangel (1992), Krishnamoorthy and Mathew (2004), Harris and Chen (2006a) and Chen and Harris (2006b). The simulation results indicate that the coverage rates of all tolerance limits are mostly below the nominal confidence level of 0.95. It suggests that the tolerance limits derived under the one-way random effects model based on the assumption of normality may not suit the non-normal distributed data.
論文目次 目錄
誌謝 i
中文摘要 ii
英文提要 iii
目錄 iv
1 緒論 1
1.1 容許區間與容許界限的定義 1
1.2 單因子隨機效應模型 2
1.3 隨機變數的設定 2
1.4 本文架構 3
2 文獻回顧 4
2.1 常用符號使用對照說明 4
2.2 七種容許界限 5
2.2.1 Mee和Owen (1983) 的方法 5
2.2.2 Vangel (1992) 的方法 6
2.2.3 Krishnamoorthy和Mathew (2004) 的方法 7
2.2.4 Chen和Harris (2006a) 的方法 9
2.2.5 Chen和Harris (2006b) 的方法 10
3 蒙地卡羅模擬研究分析 11
3.1 生成隨機變數 11
3.2 結果觀察 12
4 結論與未來研究方向 16
4.1 結論 16
4.2 未來研究 17
4.2.1 母體平均數 與 17
4.2.2 參數的估計 18
參考文獻 21
附表 23
附錄 51
參考文獻 參考文獻
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[2] Bain, L. J., Engelhardt, M. and Shiue W. K. (1984) , “Approximate Tolerance Limits and Confidence Limits on Reliability for the Gamma Distribution,” IEEE Transactions On Reliability, 33, 184–187.
[3] Chen, S. Y. and Harris, B. (2006a), “On Lower Tolerance Limits with Accurate Coverage Probabilities for the Normal Random Effects Model,” Journal of the American Statistical Association, 101, 1039–1049.
[4] Chen, S. Y. and Harris, B. (2006b), “Alternative Lower Tolerance Limits for the Normal Random Effects Model,” Unpublished Manuscript.
[5] Greenwood, J. A. and Durand, D. (1960) , “Aids for Fitting the Gamma Distribution by Maximum Likelihood,” Technometrics, 2, 55–65.
[6] Hahn, G. J. and Meeker, W. Q. (1991), “Statistical Intervals: A Guide for Practitioners,” John Wiley & Sons.
[7] Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994), “Continuous Univariate Distributions,” John Wiley & Sons, New York.
[8] Krishnamoorthy, K. and Mathew, T. (2004), “One-Sided Tolerance Limits in Balanced and Unbalanced One-Way Random Models Based on Generalized Confidence Intervals,” Technometrics, 46, 44–52.
[9] Krishnamoorthy, K. and Mathew, T. (2008), “Normal-Based Methods for a Gamma Distribution: Prediction and Tolerance Intervals and Stress-Strength Reliability,” Technometrics, 50, 69–78.
[10] Lemon, G. H. (1977), “Factors for One-Sided Tolerance Limits for Balanced One-Way-ANOVA Random-Effects Model,” Journal of the American Statistical Association, 72, 676–680.
[11] Mee, R. W. and Owen, D. B. (1983), “Improved Factors for One-Sided Tolerance Limits for Balanced One-Way ANOVA Random Model's,” Journal of the American Statistical Association, 78, 901–905.
[12] Satterthwaite, F. E. (1946), “An Approximate Distribution of Estimate of Variance Components,” Biometrics Bulletin, 2, 110––114.
[13] Stacy, E. W. (1962) , “A Generalization of the Gamma Distribution,” The Annals of Mathematical Statistics, 33, 1187––1192.
[14] Trickett, W. H. and Welch, B. L. (1954), “On the Comparison of Two Means: Further Discussion of Iterative Methods for Calculating Tables,” Biometrika, 41, 361–374.
[15] Vangel, M. G. (1992), “New Methods for One-Sided Tolerance Limits for a One-Way Balanced Random-Effects ANOVA Model,” Technometrics, 34, 176–185.
[16] Weerahandi, S. (1993), “Generalized Confidence Interval,” Journal of the American Statistical Association, 88, 899–905.
[17] Welch, B.L. (1947), “The Generalization of Student's Problem When Several Different Population Variances Are Involved,” Biometrika, 34, 28–35.
[18] Wilks, S. S. (1941), “Determination of Sample Sizes for Setting Tolerance Limits,” Annals of Mathematical Statistics, 12, 91–96.
[19] Wilson, E. B. and Hilferty, M. M. (1931) , “The Distribution of Chi-Squares, ” Proceedings of the National Academy of Sciences, 17, 684–688.
[20] 王志遠 (2007), “單因子隨機效應模型下的替代容許界限,” 淡江大學數學學系碩士論文.
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