廖振鐸臺灣大學：農藝學研究所林彩玉Lin, Tsai-YuTsai-YuLin2007-11-282018-07-112007-11-282018-07-112005http://ntur.lib.ntu.edu.tw//handle/246246/59153容許區間是一個非常重要的統計工具且被廣泛地運用於各種領域中，例如，在動植物育種研究上、生態環境品質的監控、製藥工業、或工業上製程的可靠度研究及產品品質的管控等等。容許區間是屬於統計區間的一種，其所關心的問題是母群體中的某一特定比例(proportion)個體之區間範圍。本論文主要在於發展一套容許區間的建構程序，對象乃針對包含一組相互獨立的卡方分配隨機變數的一般常態線性模型之單尾與雙尾的$\beta$-content 及 $\beta$-expectation 之容許區間。此建構程序是基於廣義基準量 (generalized pivotal quantity) 的想法來進行，原本廣義基準量的目的在解決傳統信賴區間定義下無法獲得精確解 (exact solution) 的問題。本研究所討論的統計模型，包括有一般的平衡混合線性模型 general balanced mixed linear models)、 非平衡單向隨機模型 (unbalanced one-way random models)，以及異質性 (heterogeneous)誤差變異數前提假設下具共變數 (covariates) 的非平衡單向隨機模型。就單尾的情況來說，這是一個精確的方法；就雙尾而言，則為一個好的近似方法。再由建構的程序中可以發現，廣義基準量的使用對於我們所感興趣的容許區間之建構相當簡易直接。我們由一些實際的問題來說明本論文所提出的容許區間之方法及應用。更進一步，我們進行統計模擬研究來評估此程序之成效。由模擬的結果顯示，本論文所提出的容許區間之建構程序可被建議使用於一般實際問題的解決上。A tolerance interval, a statistical interval pertains to a specified proportion of a population, is an important statistical tool and widely used in various practical applications, such as plant or animal inbreeding, environmental monitoring and regulation, pharmaceutical engineering, process reliability studies, quality control, etc. In this dissertation, we develop procedures for one- and two-sided $ eta$-content and $ eta$-expectation tolerance intervals for normal general linear models in which there exists a set of independent scaled chi-squared random variables. The developed procedures are based on the concept of generalized pivotal quantities, which has been frequently used to obtain confidence intervals in ituations where standard procedures do not lead to useful solutions. We first derive the tolerance intervals for a general setting. Then we implement the derived procedures in the general balanced mixed linear models, the unbalanced one-way random models, and the unbalanced one-way random models with covariates under heterogeneous error variances. For the one-sided case, it does not involve any approximations, resulting in an exact method. For the two-sided case, it is good approximate. It is shown that the use of generalized pivotal quantities allows the construction of the tolerance intervals of interest fairly straightforward. Some practical examples are given to illustrate the proposed procedures. Furthermore, detailed statistical simulation studies are conducted to evaluate their performance, showing that the proposed procedures can be recommended for practical use.Contents 1 Introduction 1 1.1 Applications of tolerance intervals . . . . . . . . . . . . . . . . . . . 2 1.2 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Organization of the dissertation . . . . . . . . . . . . . . . . . . . . 8 2 Preliminaries 9 2.1 Statistical intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Tolerance intervals . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Prediction intervals . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.3 Some relationships . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Generalized con dence intervals . . . . . . . . . . . . . . . . . . . . 12 3 Derivation of Tolerance Intervals for a General Setting 15 3.1 -content tolerance intervals . . . . . . . . . . . . . . . . . . . . . . 16 3.2 -expectation tolerance intervals . . . . . . . . . . . . . . . . . . . . 18 4 Tolerance Intervals for General Balanced Mixed Linear Models 20 4.1 GPQs for general balanced mixed models . . . . . . . . . . . . . . . 20 4.2 -content tolerance intervals . . . . . . . . . . . . . . . . . . . . . . 22 4.2.1 Monte-Carlo algorithm . . . . . . . . . . . . . . . . . . . . . 22 4.2.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 24 4.2.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 -expectation tolerance intervals . . . . . . . . . . . . . . . . . . . . 35 4.3.1 Monte-Carlo algorithm . . . . . . . . . . . . . . . . . . . . . 35 4.3.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 36 4.3.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 38 4.3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5 Tolerance Intervals for Unbalanced One-Way Random Models 51 5.1 A canonical form (LaMotte and McWhorter, 1978) . . . . . . . . . 52 5.2 The distributions of interest . . . . . . . . . . . . . . . . . . . . . . 53 5.3 -content tolerance intervals . . . . . . . . . . . . . . . . . . . . . . 54 5.3.1 Monte-Carlo sampling for the required GPQs . . . . . . . . 54 5.3.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 56 5.3.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3.5 Proof of Equation (5.5) . . . . . . . . . . . . . . . . . . . . . 75 5.4 -expectation tolerance intervals . . . . . . . . . . . . . . . . . . . . 77 5.4.1 Monte-Carlo sampling for the required GPQs . . . . . . . . 77 5.4.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 77 5.4.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 78 6 Tolerance Intervals for Unbalanced One-Way Random Models with Covariates under Heterogeneous Error Variances 89 6.1 The distribution of interest . . . . . . . . . . . . . . . . . . . . . . . 90 6.2 -content tolerance intervals . . . . . . . . . . . . . . . . . . . . . . 91 6.2.1 Monte-Carlo sampling for the required GPQs . . . . . . . . 91 6.2.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 94 6.2.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 98 6.3 -expectation tolerance intervals . . . . . . . . . . . . . . . . . . . . 103 6.3.1 Monte-Carlo sampling for the required GPQs . . . . . . . . 103 6.3.2 Illustrative examples . . . . . . . . . . . . . . . . . . . . . . 103 6.3.3 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . 104 7 Conclusions and Future Research 108 Bibliography 111 List of Tables 4.1 Sires' breeding values and their daughters' milk production (kg). . . 29 4.2 Simulated con dence coe cients (times 104) for the ( = 0:95; = 0:90)-tolerance intervals, based on the glucose monitoring meter experiment, with m = 5; 10; B = L = E = 3 and R = 1. . . . . . . . 32 4.3 Simulated con dence coe cients (times 104) for the ( = 0:95; = 0:90)-tolerance intervals, based on the glucose monitoring meter experiment, with m = 25; 50; B = L = E = 3 and R = 1. . . . . . . . 33 4.4 Simulated con dence coe cients (times 104) for the ( = 0:95; = 0:90)-tolerance intervals, based on the glucose monitoring meter experiment, with m = 75; 100; B = L = E = 3 and R = 1. . . . . . . 34 4.5 Breaking strength (pounds tension) of nine batches of cement briquettes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.6 The average and standard deviation (in parentheses) of simulated proportions for one-sided -expectation tolerance limits based on the glucose monitoring meter experiment for B = L = E = 3, R = 1 and T = :5. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.7 The average and standard deviation (in parentheses) of simulated proportions for one-sided -expectation tolerance limits based on the glucose monitoring meter experiment for B = L = E = 3, R = 1 and T = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.8 The average and standard deviation (in parentheses) of simulated proportions for one-sided -expectation tolerance limits based on the glucose monitoring meter experiment for B = L = E = 3, R = 1 and T = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.9 The average and standard deviation (in parentheses) of simulated proportions for one-sided -expectation tolerance limits based on the glucose monitoring meter experiment for B = L = E = 3, R = 1 and T = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.10 The average of simulated proportions for two-sided -expectation tolerance intervals using Mee's method and the proposed method. The simulated expected lengths are given in parentheses. . . . . . . 49 5.1 Sulfur content for SRM 2682. . . . . . . . . . . . . . . . . . . . . . 57 5.2 Net weights (oz) of vegetable oil lls by groups. . . . . . . . . . . . 58 5.3 Moisture contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4 Designs used for simulation study. . . . . . . . . . . . . . . . . . . . 61 5.5 Simulated con dence coe cients (times 104) and expected values for upper ( = 0:90; = 0:95)-tolerance limits of N( ; 2 ). . . . . . . . 63 5.6 Simulated con dence coe cients (times 104) and expected lengths for two-sided ( = 0:90; = 0:95)-tolerance intervals of N( ; 2 ). . . 65 5.7 Simulated con dence coe cients (times 104) and expected values for upper ( = 0:90; = 0:95)-tolerance limits of N( ; 2 + 2 e ). . . . . 67 5.8 Simulated con dence coe cients (times 104) and expected lengths for two-sided ( = 0:90; = 0:95)-tolerance intervals of N( ; 2 + 2 e ). 69 5.9 Simulated con dence coe cients (times 104) and expected values for upper ( = 0:90; = 0:95)-tolerance limits of N( ; 2 ) using Krishnamoorthy-Mathew (K-M) method (the generalized -con dence upper bound using T4 in (20)) and the proposed method. . . . . . . 73 5.10 Simulated con dence coe cients (times 104) and expected values for upper ( = 0:90; = 0:95)-tolerance limits of N( ; 2 + 2 e ) using Krishnamoorthy-Mathew (K-M) method (the generalized - con dence upper bound using T2 in (13)) and the proposed method. 74 5.11 The average and standard deviation (in parentheses) of simulated proportions for one-sided ( = 0:05)-expectation tolerance limits of N( ; 2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.12 The average and standard deviation (in parentheses) of simulated proportions for one-sided ( = 0:95)-expectation tolerance limits of N( ; 2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.13 The average and standard deviation (in parentheses) of simulated proportions for one-sided ( = 0:05)-expectation tolerance limits of N( ; 2 + 2 e ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.14 The average and standard deviation (in parentheses) of simulated proportions for one-sided ( = 0:95)-expectation tolerance limits of N( ; 2 + 2 e ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 6.1 Arsenic in NIST SRM 1556a (Oyster Tissue). . . . . . . . . . . . . 95 6.2 Number of participants, intake and steady state serum concentration of the vitamin C studies. . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 The estimated true lower serum levels (lower ( = 0:90; = 0:95)- tolerance limits) for some speci ed intake doses. . . . . . . . . . . . 97 6.4 Sampling designs used in the simulation study. . . . . . . . . . . . . 99 6.5 Simulated con dence coe cients (times 104) for one-sided ( = 0:90; = 0:95)-tolerance interval of N(x00 ; 2 ). . . . . . . . . . . . 101 6.6 Simulated con dence coe cients (times 104) for two-sided ( = 0:90; = 0:95)-tolerance interval of N(x00 ; 2 ). . . . . . . . . . . . 102 6.7 The estimated true lower serum levels (lower ( = 0:90)-expectation tolerance limits) for some speci ed intake doses. . . . . . . . . . . . 104 6.8 The average of simulated proportions for one-sided ( = 0:05)- expectation tolerance limits of N(x00 ; 2 ). . . . . . . . . . . . . . . 105 6.9 The average of simulated proportions for one-sided ( = 0:95)- expectation tolerance limits of N(x00 ; 2 ). . . . . . . . . . . . . . . 106 List of Figures 6.1 Mean serum level of vitamin C (mg/dl) versus daily intake in mg listed in Table 6.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.2 The estimated true lower ( = 0:90; = 0:95)-tolerance limits versus some speci ed daily intake in mg. . . . . . . . . . . . . . . . . . . . 98805902 bytesapplication/pdfen-US廣義 p 值線性模型卡方近似變異數成份廣義信賴區間Chi-squared approximationGeneralized p-valueGeneralized confidence intervalLinear modelVariance componentthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/59153/1/ntu-94-D89621201-1.pdf