多變量良質率之區間估計

Abstract

工業產品的產製過程，對於產品的品質特徵值(quality characteristic)的要求通常是期望高過所期望的一定高之比例的產品落入預定的規格(specification acceptance region)內，符合此要求的產品之比例(proportion)，即是所謂的良質率(conformance proportion)。對應到統計學上，即是以一個隨機變數(random variable)來代表此一品質特徵值，此隨機變數落於某個給定的規格範圍(specification limits)的比例即為良質率。由於實際產製過程，產品的品質往往是由多個品質特徵值(multiple quality characteristics)決定，所以本論文的研究著重於多變量下之良質率的區間估計。我們提出兩類型的估計方法，第一類型是依照廣義樞紐量(Fiducial Generalized Pivotal Quantity)(FGPQ)的概念推導出的方法(FGPQ-based method)，另一類型則是利用二項分佈推導出的方法(Binominal-distribution-based method)。本論文的第二章討論單一個多變量常態分佈(multivariate normal distribution)之良質率的區間估計，此處我們建構信賴下界(lower confidence limit)，對此良質率進行右尾假設檢定。第三章則是建構兩個良質率之差異(difference)的信賴區間(two-sided confidence interval) 以進行雙尾假設檢定。我們利用統計模擬來評估各個方法的表現，實際應用時，若無給定的規格範圍，則使用容許區間(statistical tolerance intervals)去建構適當的規格範圍，除了與製程良率相關的實例外，本論文亦討論台灣地區空氣品質及物種判別的實際資料分析。

An industrial product is considered to be actually usable, but not scrap or defective, it usually needs to meet multiple performance requirements, which are referred to as quality characteristics. The individual component of such quality characteristics may be qualitative (nominal or ordinal) or quantitative (discrete or continuous). In specificity, the performance requirement of each single quantitative characteristic is often specified by a target value along with a conformance region or an acceptance region, which is described by means of a lower and an upper specification limit. A product is deemed to be a conforming product if all of the quality characteristics fall into these specification limits. Confidence limits for the conformance proportion are usually required not only to perform statistical significance test, but also to provide useful information for determining practical significance. In this dissertation, we develop approaches for computing confidence limits for the conformance proportion of multiple quality characteristics that are assumed to be distributed as a multivariate normal distribution. Based on the concept of a fiducial generalized pivotal quantity (FGPQ) and on the Binomial distribution by treating it as the success proportion of a binary population. We first focus on a single conformance proportion, and then extend to the situation of a difference between two the conformance proportions. The estimation performance of the proposed methods is evaluated through detailed simulation studies. Some real data examples are collected to illustrate the application of the conformance proportion, including manufacture process data, air quality data and flea beetle species data, etc.