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研究生:江怡蓁
研究生(外文):Yi-Chen Chiang
論文名稱:條件最大概似估計在相關資料之管制圖的應用
論文名稱(外文):Conditional Maximum Likelihood Estimation for Control Charts in the Presence of Correlation
指導教授:蔡宗儒蔡宗儒引用關係
指導教授(外文):Tzong-Ru Tsai
學位類別:碩士
校院名稱:淡江大學
系所名稱:統計學系
學門:商業及管理學門
學類:會計學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:中文
論文頁數:43
中文關鍵詞:自我迴歸移動平均模型一階自我迴歸模型EWMA管制圖史瓦特管制圖最大概似估計
外文關鍵詞:Autoregressive Moving Average ModelFirst-order Autoregressive ModelExponentially Weighted Moving Average Control ChartsShwehart Control ChartMaximum Likelihood Estimation
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管制圖被廣泛的應用於製程監控,當製程失控時,可以及時發現,減少損失,因此,如何建構一個適當的管制圖是十分重要的。理論上,在建構管制圖時,我們通常假設觀察值之間彼此獨立,但實際上觀察值之間卻常存在著相關性,彼此互相影響。Reynolds et al [6]指出根據觀察值獨立的基本假設來建構管制圖,在很多情況下,很容易產生錯誤的製程監控,造成在管制狀態下(in control)的平均連串長度(Average Run Length,簡寫為ARL)降低,並使得錯誤警訊率(false alarm rate)高於當觀察值獨立時的製程。像這樣的製程監控不但無法提高產品的品質、降低成本及節省時間,反而會造成許多浪費。為了適當的監控製程平均,當觀察值來自AR(1)加上隨機誤差模型(AR(1)+random error model)時,本文利用條件最大概似估計法估計模型參數,以克服在小樣本下,容易造成估計不準確的問題,並使得所建構的管制圖有較良好的使用成效。此外,本文也提出在小樣本之下,不同相關程度(由小至中)的參數估計結果,並與真實資料作比較。
The control chart is extensively applied in monitoring process, it can detect in time to decrease the loss when the process is out of control, therefore, how to construct a appropriate control chart is very important. Theoretically, when we construct a control chart we assume observations are independent, but in practical applications, the observations in many cases are frequently autocorrelated.
The autocorrelation between successive observations has a large impact on the control charts developed under the independent assumption; Reynolds et al[6] point out that it can decrease the in control average run length(ARL) and produce a higher false alarm rate than for an independent process. Such kind of control chart is not only unable to promote the quality of the products, lower the costs, but also makes many wastes. Consequently, we research how to construct suitable control charts in the situation that the observations are autocorrelated.
We consider the problem of monitoring the mean of AR(1) process with random error model and provide a conditional maximum likelihood estimation method to improve the chart performance when the sample size is small. There is a large probability resulting that the standard estimation method breaks down if the level of correlation between successive means are small-to-moderate and the sample size is small. Besides, we also provide numerical results under estimating the parameter of different levels of correlation, and compare them with the true data.
目錄
第一章 緒論...............................................1
1.1 前言.................................................1
1.2 研究動機與目的.......................................1
1.3 相關研究與文獻探討...................................3
1.4 本文主要概念.........................................4
1.5 本文架構.............................................5
第二章 Reynolds et al概念與方法介紹.......................6
2.1 前言.................................................6
2.2 基本概念與方法.......................................6
2.3 Reynolds et al方法的設限............................10
2.4 標準概似估計方法模擬實例............................11
第三章 條件最大概似估計法................................16
3.1 前言................................................16
3.2 條件最大概似估計法..................................16
第四章 模擬與數值實例運用
4.1 前言................................................22
4.2 兩個相關的管制圖....................................22
4.3 數值實例............................................24
4.4 模擬結果............................................26
第五章 結論與建議.........................................28
5.1 總結................................................28
5.2 後續發展............................................29
~
附錄A μ的展開式...........................................40
參考文獻...................................................42
表目錄
表1 n=100時,Bias、MSE、Pr-1的模擬結果....................30
表2 n=300時,Bias、MSE、Pr-1的模擬結果....................30
表3 n=500時,Bias、MSE、Pr-1的模擬結果....................31
表4 Ψ=0.5時,EWMA管制圖c與λ的選擇表.....................32
表5 Ψ=0.9時,EWMA管制圖c與λ的選擇表.....................33
表6 史瓦特管制圖h值的選擇表...............................34
表7 三種不同方法的估計結果................................34
表8 根據條件最大概似估計法估計所得之結果..................35
圖目錄
圖1 估計Φ值之Bias走勢圖..................................36
圖2 估計Θ值之Bias走勢圖..................................36
圖3 估計Φ值之MSE走勢圖...................................37
圖4 估計Θ值之MSE走勢圖...................................37
圖5 不同Φ值下之Pr-1走勢圖................................38
圖6 120個模擬值的時間序列圖...............................38
圖7 根據模擬值所建構的史瓦特管制圖........................39
圖8 根據模擬值所建構的EWMA管制圖..........................39
[1] Abraham, B. and Kartha, C. P. (1979). Forecast Stability and Control Charts.ASQC Technical Conference Transactions. American Society for Quality Control, Milwaukee, WI. pp. 675-680.
[2] Alwan, L. C. (1991). Autocorrelation: Fixed Versus Variable Control Limits.Quality Engineering, 4, pp. 167-188.
[3] Alwan, L. C. and Roberts, H. V. (1988). Time-Series Modeling for Statistical Process Control. Journal of Business and Economic Statistics, 6, pp. 87-95.
[4] Box, G. E. P., Jenkins, G. M. and Reinsel, G. C. (1994). Times series Analysis, Forecasting and Control, 3rd ed. Prentice-Hall, Englewood Cliffs, New Jersey.
[5] Harris, T. J. and Ross, W. H. (1991). Statistical Process Control Procedures for Correlated Observations. The Canadian Journal of Chemical Engineering, 69, pp. 48-57.
[6] Lu, C-W. and Reynolds, M. R., JR. (1999a). EWMA Control Charts for Monitoring the Mean of Autocorrelated Process. Journal of Quality Technology, 31. pp. 166-188.
[7] Lu, C-W. and Reynolds, M. R., JR. (1999b). Control Charts for Monitoring the Mean and Variance of Autocorrelated Process.
Journal of Quality Technology, 31. pp. 259-274.
[8] Lu, C-W. and Reynolds, M. R., JR. (2001). CUSUM Charts for Monitoring an Autocorrelated Process. Journal of Quality Technology, 33. pp. 316-334.
[9] Montgomery, D. C. (2000). Introduction to Statistical
Quality Control, 4th ed. John Wiley & Sons, New York, NY.
[10] Reynolds, M. R., JR., Arnold, J. C. and Baik, J. W. (1996). Variable Sampling Interval Charts in the Presence of Correlation. Journal of Quality Technology, 28. pp. 12-30.
[11] VanBrackle, L. N. and Reynolds, M. R., JR. (1997). EWMA and CUSUM Control charts in the Presence of Correlation.
Communications in Statistics-Simulation and Computation, 26.pp.
979-1008.
[12] Vasilopoulos, A. V. and Stamboulis, A. P. (1978). Modification of Control Limits in the Presence of Data Correlation. Journal of Quality Technology, 10. pp. 20-30.
[13] Wardell, D. G., Moskowitz, H. and Plante, R. D. (1994). Run Length Distributions of Special-Cause Control Charts for Correlated Processes. Technometrics, 36. pp. 3-17.
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