本論文針對平衡的雙因子套層隨機效應模型 (two-way nested 
random effects model) 推導出單邊容許界限。我們藉由 Chen 和
 Harris(2006) 所提出的方法，即在估計量為未知期望均方比值
 (unknown expected mean square ratio)的條件下，求條件機率密
 度函數，而所找到的統計量與均方比值獨立且具有最大自由度的卡
 方分配，並以此統計量建立一個精準 (exact)的單邊容許界限。利
 用統計模擬來驗證此方法的可行性，其結果顯示在此模型下亦有準
 確的模擬涵蓋率及較小的平均值與標準差，因此經數值探討後發現
 本方法比其他方法不僅不保守且更準確。最後舉出三個實際的例子
 來做方法比較，並提出利用此方法所需的表格。

We consider in this article a method of constructing accurate  $beta$-content tolerance limits for a two-way nested model with normal random effects. The procedure is derived by conditioning on an estimator of the unknown expected mean square ratio as proposed in Chen and Harris (2006). Simulation studies indicate that the present procedure is less conservative than current existing methods and gives more accurate coverage rates. Statistical tables needed to implement the procedure are included.

目錄
1. 緒論 1
1.1 單邊容許界限的定義 2
2 文獻回顧 5
2.1 單因子隨機效應模型 5
2.2.1 Chen 和 Harris (2006) 的方法 6
2.2 雙因子套層隨機效應模型 7
2.2.2 Fonseca et al. (2007) 的方法 8
3. 平衡的隨機效應模型下的容許界限 11
3.1 容許上界的模擬研究 15
3.2 線性內插法求 (η_1,η_2) 值 17
4. 特殊變異數比值的結果 18
4.1 當 R_1 與 R_2 值皆為零的情形 18
4.2 當 R_1 與 R_2 其中一值為零的情形 19
4.2.1 當 R_1 值為零的情形 19
4.2.2 當 R_2 值為零的情形 20
4.3 當 R_1 與 R_2 其中一值為趨近無限大的情形 21
4.3.1 當 R_2→∞ 的情形 22
4.3.2 當 R_1→∞ 的情形 22
5. 容許界限的模擬比較 25
5.1 模擬研究 (一)  25
5.2 模擬研究 (二)  31
6. 實例分析 36
6.1 實例一 36
6.2 實例二 38
6.2.1 實例二中 CL 的查表方法 39
6.3 實例三 40
7. 結論 42
附表 A.1 (η_1,η_2) values for (P,γ)=(0.90,0.95) and n=2 43
附表 A.2 (η_1,η_2) values for (P,γ)=(0.90,0.95) and n=5 49
附表 A.3 (η_1,η_2) values for (P,γ)=(0.90,0.95) and n=20 55
附錄 A 引理 61
附錄 B Fortran程式 64
參考文獻 74
表目錄
1. 單因子隨機效應模型的變異數分析表 6
2. 雙因子套層隨機效應模型的變異數分析表 8
3. 當 R_1=0 時，雙因子套層隨機效應模型縮減成單因子隨機效應模型的變異數分析表 19
4. 當 R_2=0 時，雙因子套層隨機效應模型縮減成單因子隨機效應模型的變異數分析表 21
5. 當 (P,γ)=(0.90,0.95), a=5, b=5, n=20 及 R_2=1 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 27
6. 當 (P,γ)=(0.90,0.95), a=5, b=20, n=20 及 R_2=1 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 28
7. 當 (P,γ)=(0.90,0.95), a=20, b=5, n=20 及 R_2=1 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 29
8. 當 (P,γ)=(0.90,0.95), a=20, b=20, n=20 及 R_2=1 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 30
9. 當 (P,γ)=(0.90,0.95), a=3, b=3, n=5 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 32
10. 當 (P,γ)=(0.90,0.95), a=3, b=10, n=5時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 33
11. 當 (P,γ)=(0.90,0.95), a=10, b=3, n=5 時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 34
12. 當 (P,γ)=(0.90,0.95), a=10, b=10, n=5時，模擬估計出三種單邊容許上界的涵蓋率，平均值及標準差 35
13. 每隻小豬平均增加的體重 36
14. 每隻小豬平均增加的體重之變異數分析表 37
15. 蕪菁葉子的鈣濃度資料 (%乾重) 38
16. 蕪菁葉子的鈣濃度資料之變異數分析表 38
17. 量具量測所得的數據 40
18. 量具量測所得的數據之變異數分析表 41

Aspin, A. A. (1948), “An Examination and Further Development of a Formula Arising in the Problem of Comparing Two Mean Values,” Biometrika, 35, 89–96.

Bhaumik, D. K., and Kulkarni, P. M. (1996), “A Simple and Exact Method of Constructing Tolerance Intervals for the One-Way ANOVA with Random Effects,”
The American Statistician, 50, 319–323.

Chen, S. Y. and Harris, B. (2006), “On Lower Tolerance Limits with Accurate Coverage Probabilities for the Normal Random Effects Model,” Journal of the American Statistical Association, 101, 1039–1049.

Fonseca, M., Mathew, T., and Mexia, J. T., Zmy´slony, R. (2007), “Tolerance Intervals in a Two-Way Nested Model with Mixed or Random Effects,” Statistics, 41(4), 289–300.

Hahn, G. J., and Meeker, W. Q. (1991), Statistical Intervals: A Guide for Practitioners, John Wiley & Sons.

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(1), 44–52.

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.

Mee, R. W., and Owen, D. B. (1983), “Improved Factors for One-Sided Tolerance Limits for Balanced One-Way ANOVA Random Model,” Journal of the American Statistical Association, 78, 901–905.

MIL–HDBK–17D (1994), Polymer Matrix Composites, Volume I: Guidelines, Philadelphia: Naval Publications and Forms Center.

Owen, D. B. (1963), “Factors for One-Sided Tolerance Limits and for Variables Sampling Plans, ” Monograph SCR-607, Sandia Corporation, Albuquerque.

Sahai, H., and Ageel, M. I. (2000), The Analysis of Variance: Fixed, Random and Mixed Models, (Boston: Birkhauser).

Satterthwaite, F. E. (1946), “An Approximate Distribution of Estimate of Variance Components,” Biometrics Bulletin, 2, 110–114.

Snedecor, G. W., and Cochran, W. G. (1967), Statistical Methods, 6th ed., The Iowa State University Press, Ames, pp. 285–288.

Tamhane, A. C. (2009), Statistical Analysis of Designed Experiments: Theory and Applications, John Wiley & Sons.

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.

Vangel, M. G. (1992), “New Methods for One-Sided Tolerance Limits for a One-Way Balanced Random-Effects ANOVA Model,” Technometrics, 34, 176–185.

Weerahandi, S. (1993), “Generalized Confidence Intervals,” Journal of the American Statistical Association, 88, 899–905.

Welch, B. L. (1947), “The Generalization of Student’s Problem When Several Different Population Variances Are Involved,” Biometrika, 34, 28–35.

Wilks, S. S. (1941), “Determination of Sample Sizes for Setting Tolerance Limits,” Annals of Mathematical Statistics, 12(1), 91–96.