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研究生:粘佑任
研究生(外文):NIAN, YOU-REN
論文名稱:偏斜常態分配在線性模型中的應用
論文名稱(外文):Linear regression model with skew normal distribution
指導教授:蘇南誠蘇南誠引用關係
指導教授(外文):Su, Nan-Cheng
口試委員:張升懋蘇南誠黃佳慧
口試委員(外文):Chang, Sheng-MaoSu, Nan-ChengHuang, Chia-Hui
口試日期:2017-06-12
學位類別:碩士
校院名稱:國立臺北大學
系所名稱:統計學系
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:英文
論文頁數:31
中文關鍵詞:偏斜常態分配多變量線性迴歸模型動差估計量最小平方估計量
外文關鍵詞:Skew normal distributionMultivariatelinear regression modelLeast squares estimatorMethod of moment estimator
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由於在蒐集資料時,儘管會盡量要求每個樣本之間是相互獨立的情況,但實際上,樣本之間卻是仍有相互影響的情況。鑒於上述情況,原本建立在誤差項服從獨立且分布相同的常態假設的線性迴歸模型將不合適,因為沒有考慮到樣本之間相互影響的情況,又或是不再服從常態分配。因此我們考慮在不同分配假設下去探討線性迴歸的狀況,包含推導動差估計量、最小平方估計量以及最大概似估計量,以及其所相對應的分佈情況。其中,我們將考慮線性迴歸模型中誤差項的分配假設為多變量偏斜常態分配,或是每個誤差項均相互獨立且服從相同的單維度偏斜常態分配。接著,我們使用兩筆資料來觀察在不同假設下的迴歸模型配適的情況。最後,我們將討論線性迴歸模型延伸至線性混合模型對於隨機效應項和誤差項使用偏斜常態假設時的情況。
It is always the question that the data is independent. When the error terms is not independently following the normal distribution, the traditional linear regression model is not the appropriate model in the data. Considering the situation of data is dependent or data is no loner following the normal distribution, we try to use the skew normal distribution to replace the normal distribution as the assumption of the error. Therefore, we calculate the method of moment estimator, least squares estimator and maximum likelihood estimator. Besides, we try using the above estimators to fit two data. In the end, we extend the linear regression model to the linear mixed model.
Contents
1 Introduction 1
2 Linear Regression Model 3
2.1 Method of moment estimate (MME) . . . . . . . . . . 3
2.2 Method of Least Squares Estimate . . . . . . . . . . . 5
3 Skew-Normal Distribution 7
3.1 The errors is correlated . . . . . . . . . . . . . . . . . 7
3.2 The errors is independent . . . . . . . . . . . . . . . . 9
4 Simulation study 11
4.1 Normal case . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Skew-normal case . . . . . . . . . . . . . . . . . . . . 13
4.3 Independent Skew-normal case . . . . . . . . . . . . . 15
5 Example 23
5.1 Example 1: House prices in Iowa . . . . . . . . . . . . 23
5.2 Example 2: University Admissions . . . . . . . . . . . 24
6 Conclusion 28
Reference 30

List of Figures
4.1 n = 50, ε ~iid N(0, 9), β= (1, 0.01)' . . . . . . . . . . . 13
4.2 n = 50, ε ~iid N(0, 9); β = (0.01, 0.01)' . . . . . . . . . 14
4.3 n = 100, ε~SN_100(0, 9 x I_100, 1), β = (1,2)' . . . . . 16
4.4 n = 100, ε~SN_100(0, 9 x I_100, 1), β = (1, 0.01)' . . . 17
4.5 n = 20, ε~iid SN(0, 3, 1), β = (0.01, 0.01)' . . . . . . . 18
4.6 n = 20, ε~iid SN(0, 3, 1), β = (1, 2)' . . . . . . . . . . 19
4.7 n = 50, ε~iid SN(0, 3, 1), β = (0.01, 0.01)' . . . . . . . 19
4.8 n = 50, ε~iid SN(0, 3, 1), β = (1, 2)' . . . . . . . . . . 20
4.9 n = 100, ε~iid SN(0, 3, 1), β = (0.01, 0.01)' . . . . . . 20
4.10 n = 100, ε~iid SN(0, 3, 1), β = (1, 2)' . . . . . . . . . . 21
4.11 n = 300, ε~iid SN(0, 3, 1), β = (0.01, 0.01)' . . . . . . 21
4.12 n = 300, ε~iid SN(0, 3, 1), β = (1, 2)' . . . . . . . . . . 22
5.1 Diagnostic for residuals at house data . . . . . . . . . 26
5.2 Diagnostic for residuals at GPA data . . . . . . . . . 27

List of Tables
4.1 ε ~iid N(0, 9), β = (1, 2)' . . . . . . . . . . . . . . . . . 12
4.2 ε ~iid N(0, 9), β = (1, 0.01)' . . . . . . . . . . . . . . . 12
4.3 ANOVA test on simulation data with ε ~iid N(0, 9), β = (1; 2)' , n = 300 . . . . . . . . . . . . . . . . . . . . . 12
4.4 ANOVA test on simulation data with ε ~iid N(0, 9), β = (1, 0.01)' , n = 300 . . . . . . . . . . . . . . . . . . . 13
4.5 ε ~ SN_100(0, 9 x I_100, 1), β = (1, 2)' . . . . . . . . . . 14
4.6 ε ~ SN_100(0, 9 x I_100, 1), β = (1, 0.01)' . . . . . . . . 15
4.7 ANOVA test on simulation data with ε ~ SN_100(0. 9 x I_100, 1), β = (1, 2)' , n = 300 . . . . . . . . . . . . . . 15
4.8 ANOVA test on simulation data with ε ~ SN_100(0, 9 x I_100, 1), β = (1, 0.01)' , n = 300 . . . . . . . . . . . . . 16
4.9 ε ~iid SN(0, 3, 1), β = (1, 2)' . . . . . . . . . . . . . . . 17
4.10 ε ~iid SN(0, 3, 1), β = (1, 0.01)' . . . . . . . . . . . . . 17
4.11 ANOVA test on simulation data with ε ~iid SN(0, 3, 1), β = (1; 2)' , n = 300 . . . . . . . . . . . . . . . . . . . . . 18
4.12 ANOVA test on simulation data with ε ~iid SN(0, 3, 1), β = (1, 0.01)' ; n = 300 . . . . . . . . . . . . . . . . . . . 18
5.1 Parameter estimates of house prices data. . . . . . . . 24
5.2 Parameter estimates of university admissions data. . . 26
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