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論文名稱(中文) 偏斜常態均勻分佈及其應用
論文名稱(英文) A Study of Skew Normal-Uniform Model and its Applications
校院名稱 成功大學
系所名稱(中) 統計學系碩博士班
系所名稱(英) Department of Statistics
學年度 100
學期 1
出版年 101
研究生(中文) 徐政益
研究生(英文) Cheng-Yi Hsu
學號 R26981083
學位類別 碩士
語文別 英文
論文頁數 32頁
口試委員 指導教授-蘇南誠
口試委員-張升懋
口試委員-蘇志成
中文關鍵字 偏斜分佈  偏斜常態均勻分佈  平均數管制圖  全距管制圖 
英文關鍵字 Skewed distribution  skew-normal-uniform distribution  X-bar control chart  R control chart 
學科別分類
中文摘要 研究機率分佈的性質一直以來都是統計和應用機率領域的重要課題。Azzalini (1985) 提出的偏斜常態分佈,不僅涵蓋常態分佈,且具有一些與常態分佈相同的性質。此類分佈有助於穩健性的研究和偏斜性的建模。此後,便有許多人投入基於對稱分佈的偏斜分佈之研究。本論文將完整地探討所謂的偏斜常態均勻分佈,除了基本性質外,也將探討其點估計和平均數之分佈。偏斜常態均勻分佈不但可比典型且被廣泛討論的常態分佈更具彈性,亦提供不同於Azzalini之偏斜常態分佈的偏斜模型選擇。在應用方面,我們將探討此偏斜常態均勻分佈在品質管制圖的制定上,以期提供一個可用且有效的統計模型來處理非對稱資料,並改善修華特管制圖無法達到監控非對稱資料的目的。
英文摘要 The study of properties of probability distributions has always been a persistent theme of statistics and applied probability. Azzalini (1985) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by more authors. In this thesis, we study the so-called skew-normal-uniform distribution, which is flexible than the normal distribution and different from Azzalini's skew-normal distribution. Explicit forms of its c.d.f. and moments are derived. The method of moments estimation and the maximum likelihood estimation of location-scale skew-normal-uniform distribution are discussed. We also study the distribution of the sample mean and then study control charts for the skew-normal-uniform distribution to monitor the process average of non-normal data. This improves the defect of Shewhart control chart and provides an effective model to deal with the non-normal data.
論文目次 1 Introduction 1
2 Basic properties of skew-normal-uniform distribution 3
3 Point estimation 12
3.1 Method of moment estimation. . . . . . . . . . . 12
3.2 Maximum likelihood estimation. . . . . . . . . . 13
4 Distribution of SNU sample mean 14
5 Control chart for SNU data 21
5.1 X-bar control chart based on maximum likelihood
estimation . . . . . . . . . . . . . . . . . . . 22
5.2 X-bar Control chart based on method of moment
estimation . . . . . . . . . . . . . . . . . . . 23
5.3 R control chart. . . . . . . . . . . . . . . . . 24
6 Numerical example 25
6.1 Example 1 . . .. . . . . . . . . . . . . . . . . . 25
6.2 Example 2 . . . . .. . . . . . . . . . . . . . . . 28
7 Conclusions and Future work 31
8 References 31
參考文獻 1. Arellano-Valle, R., G omez, H., and Quintana, F. (2004). A new class of skew-normal distributions. Communications in Statistics-Theory and Methods, 33(7):1465-1480.

2. Azzalini, A. (1985). A class of distributions which includes the normal ones. Scan-dinavian Journal of Statistics, 12:171-178.

3. Bai, D. and Choi, I. (1995). X and R control charts for skewed populations. Journal of Quality Technology, 2(2):120-131.

4. Chan, L. and Cui, H. (2003). Skewness correction X and R charts for skewed distributions. Naval Research Logistics (NRL), 50(6):555-573.

5. Chang, Y. and Bai, D. (2001). Control charts for
positively-skewed populations with weighted standard deviations. Quality and Reliability Engineering International,17(5):397-406.

6. Chen, J., Gupta, A., and Nguyen, T. (2004). The density of the skew normal sample mean and its applications. Journal of Statistical Computation and Simulation, 74(7):487-494.

7. G omez, H., Venegas, O., and Bolfarine, H. (2007). Skew-symmetric distributions generated by the distribution function of the normal distribution. Environmetrics, 18(4):395-407.

8. Gupta, A., Chang, F., and Huang, W. (2002). Some skew-symmetric models. Ran-dom Operators and Stochastic Equations, 10(2):133-140.

9. Mudholkar, G. and Hutson, A. (2000). The epsilon-skew-normal distribution for analyzingnear-normal data. Journal of Statistical Planning and Inference, 83(2):291-309.

10. Nadarajah, S. and Kotz, S. (2003). Skewed distributions generated by the normalkernel. Statistics and Probability Letters, 65(3):269-277.

11. Tsai, T. (2007). Skew normal distribution and the design of control charts for averages.International Journal of Reliability Quality and Safety Engineering, 14(1):49-63.
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