臺灣博碩士論文加值系統

良質率定義為一個感興趣的特性值(characteristic) 落於預先指定的可接受範圍(acceptance region) 內之比例。良質率不僅可應用於製造業的製程評估，而且在農業管理及環境監測等領域上，亦可用來幫助評估感興趣的特徵表現。例如一個適當的乾物質含量(dry matter content) 範圍有助於決定飼料用玉米的最佳收穫時機；水果的甜度(sweetness) 要求必須高於某個規格；亦或在農藥殘留檢測中要求毒素的濃度必須低於某個上限值等等，都是想要估計一個隨機變量(random variable) 超過某個給定界線(specification limit) 或是落於某個給定範圍(specification region) 的比例，本質上即是在估計良質率。
本論文建議在實際應用上，以良質率的區間估計作為容許區間(tolerance interval) 的替代方法。首先，我們討論良質率的區間估計與容許區間之間的關係與異同。接下來，對於雙邊良質率(bilateral conformance proportion)，我們建構了兩種信賴區間估計方法，一個是以廣義樞軸量(generalized pivotal quantity) 為
基礎，另一個是利用修正大樣本法(modified large sample method) 的概念。我們同樣提出兩種方法來建構單邊良質率(unilateral conformance proportion) 的信賴區間，第一種也是利用廣義樞軸量，第二種則是建構在學生氏t 分佈(Student’s t
distribution) 上。一個以拔靴法(bootstrap method) 為基礎的校正(calibration) 概念被應用到單邊及雙邊良質率的區間估計上，使其經驗覆蓋率(empirical coverage probability) 更接近給定的數值。同時，我們也考慮不均衡數據(unbalanced data)的估計。此外，我們藉由一些例子來說明本論文所提出的良質率區間估計方法及
其應用，並且透過統計模擬研究來評估這些方法的成效。結果顯示，在實際應用上，本論文所提出之良質率區間估計為可建議使用之方法。

Conformance proportion is defined as the proportion of a performance characteristic of interest that falls within a prespecified acceptance region. It can be used not only in manufacture industry but also in agricultural management or environmental monitoring. For instance, determining best harvest timing for forage maize under an appropriate range of dry matter content, monitoring the sweetness of fruits to
be above a lower limit, or requiring the concentration of a toxin to be below an upper limit in pesticide residue tests. It is of desire to estimate the probability that
a random variable exceeds a specification limit or falls into a specification region, which is essentially the conformance proportion.
In this dissertation, we propose the approach of a conformance proportion as an alternative to that of a tolerance interval for practical use. First, we discuss
the connections between the two approaches. Then, two methods are developed for computing confidence limits for bilateral conformance proportions, one is based on the concept of a generalized pivotal quantity and the other is based on the modified large sample method. For unilateral conformance proportions, we also propose two methods for interval estimation, the first one is also based on the concept of a generalized pivotal quantity and the second one is based on the Student’s t distribution. A bootstrap calibration approach is adapted for both bilateral and unilateral conformance proportions to have empirical coverage probability sufficiently close to the nominal level. Furthermore, we consider the situations with unbalanced data scenarios. Some examples are given to illustrate the proposed methods. The performances of these approaches are evaluated by detailed statistical simulation studies, showing that they can be recommended for practical use.

1 Introduction 1
1.1 Connections with statistical tolerance intervals . . 2
1.1.1 Statistical tolerance intervals . . . . . . . . . . 3
1.1.2 Confidence limits for conformance proportions . . . 4
1.1.3 Connections . . . . . . . . . . . . . . . . . . . . 4
1.2 Literature review and framework . . . . . . . . . . . 5
1.3 Organization of the dissertation . . . . . . . . . . 7
2 Approaches for the bilateral conformance proportion 8
2.1 The GPQ-based method for . . . . . . . . . . . . . 9
2.2 The MLS-based method for . . . . . . . . . . . . . 11
2.3 Illustrative examples . . . . . . . . . . . . . . . 14
2.4 Simulation studies . . . . . . . . . . . . . . . . . 20
2.5 Bootstrap-based calibration . . . . . . . . . . . . 29
3 Approaches for unilateral conformance proportions 33
3.1 The GPQ-based method for pi_L and pi_U. . . . . . . 34
3.2 The t-based method for pi_L and pi_U. . . . . . . . 34
3.2.1 au_2 is known . . . . . . . . . . . . . . . . . . 34
3.2.2 au_2 is unknown . . . . . . . . . . . . . . . . . 35
3.3 Simulation study . . . . . . . . . . .. . . . . . . . 36
3.4 Adjusted t-based method . . . . . . . . . . . . . . . 40
3.5 Illustrative examples . . . . . . . . . . . . . . . . 43
4 Unbalanced data scenarios 48
4.1 Estimation methods . . . . . . . . . . . . . . . . . 48
4.1.1 The MLS-based method for pi. . . . . . . . . . . . 49
4.1.2 The GPQ-based method for pi_L and pi_U. . . . . . 50
4.1.3 The t-based method for pi_L and pi_U. . . . . . . 50
4.2 Simulation studies . . . . . . . . . . . . . . . . .. 51
4.2.1 Bilateral conformance proportion. . . . . . . . . . 52
4.2.2 Unilateral conformance proportion . . . . . . . . . 57
4.3 Illustrative examples . . . . . . . . . . . . . . . . 65
4.4 Discussion. . . . . . . . . . . . . . . . . . . . . . 70
5 Conclusions and future research 74
Bibliography 77

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