非對稱規格區間之製程能力指標的統計性質

Abstract

製程能力指標 (Process Capability Index) 是一個衡量製程績效的方便工具，可以藉其指標的數值來評估製程，以了解此製程的產出，合乎預設規格的程度。近來，關於製程能力指標的研究工作，散見於統計與品管相關的文獻中。大部份的研究均著重於規格區間為對稱的情況，當產品規格區間為非對稱的情況時，是一般研究者較為忽略的。Pearn and Chen (1998) 提出能力指標 Cpk" 來處理非對稱規格區間之製程，新指標 Cpk" 為指標 Cpk 的推廣。本文首先在常態分配之假設下，推導出 Cpk" 的估計量的累積分配函數與機率密度函數，便於作區間估計、統計假設檢定等進一步的統計分析。隨之，將建構指標 Cpk" 的相同理念，應用到能力指標 Cpm 與 Cpmk，分別發展出新指標 Cpm" 與 Cpmk"。我們分別將這些新指標與文獻中現有的指標加以比較。在常態分配之假設下，我們分別推導出 Cpm" 與 Cpmk" 的估計量的累積分配函數與機率密度函數，也分別探討了 Cpm" 與 Cpmk" 的估計量的統計性質。

Process capability indices (PCIs), providing numerical measures of whether or not the ability of a manufacturing process meets a preset level of production tolerance, are considered as a practical tool in industry. Capability indices have received much interest in the statistical literature during recent years. Most research work, however, have focused on developing and investigating PCIs for processes with symmetric tolerances. There have been relatively few papers published dealing specially with the case when the tolerances are asymmetric. Pearn and Chen (1998) proposed a new generalization of index Cpk, called Cpk", to handle processes with asymmetric tolerances. In this dissertation, we derive the cumulative distribution function and the probability density function of the estimated index of Cpk" when sampling from a normal distribution. The explicit form of the distribution of the estimated indices under investigation could be viewed as an interesting and useful result in interval estimation and testing statistical hypothesis. Based on the same idea, we consider new generalizations of indices Cpm and Cpmk, called Cpm" and Cpmk", respectively. We make a comprehensive study of several proposed indices for asymmetric tolerances. We derive the explicit forms of the cumulative distribution functions and the probability density functions of the estimators of Cpm" and Cpmk" when sampling from a normal distribution to increase the utility of these capability indices. We also investigate the statistical properties of the natural estimators of Cpm" and Cpmk" assuming the process is normally distributed. Abstract List of Contents List of Figures List of Tables Chapter 1. Introduction Chapter 2. A New Generalization of Cpk 2.1 Introduction 2.2 Distribution of the Estimated Cpk" Chapter 3. A New Generalization of Cpm 3.1 Introduction ∙ 3.2 Existing Generalizations 3.3 A New Generalization Cpm" 3.4 Estimation of Cpm" and the Sampling Distribution 3.5 Conclusions Chapter 4. A New Generalization of Cpmk 4.1 Introduction 4.2 Existing Generalizations 4.3 A New Generalization Cpmk" 4.4 Estimation of Cpmk" and the Sampling Distribution 4.5 Conclusions Appendix A Appendix B References