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研究生:呂政勳
論文名稱:高維 Behrens-Fisher 問題
論文名稱(外文):High dimensional Behrens-Fisher problem
指導教授:周若珍周若珍引用關係
口試委員:史玉山鄭少為
口試日期:2011-06-22
學位類別:碩士
校院名稱:國立清華大學
系所名稱:統計學研究所
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:2011
畢業學年度:99
語文別:中文
論文頁數:27
中文關鍵詞:兩母體檢定Behrens-Fisher 問題高維度資料
相關次數:
  • 被引用被引用:1
  • 點閱點閱:221
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  • 收藏至我的研究室書目清單書目收藏:0
本論文將融合 Welch(1938) 與 Wu et al.(2006) 方法之概念, 提出
PCT-like 檢定, 使用兩種近似虛無分配作為 PCT-like 檢定量之虛無分配 ,來解決高維 Behrens-Fisher 問題。 並以模擬實驗, 比較兩者與 Yao(1965) 法、 Kim(1992) 法和 Wu et al.(2006) 法之型一誤差率及檢定力。 結果顯示, PCT-like 法之型一誤差率在多數情況接近目標值, 而檢定力皆與其他法表現相當。 本文亦對現存的 「母體變異是否相等」 檢定法作了比較。 最後以柏拉圖作品風格與大腸癌篩檢資料作為實證。
目錄
1 緒論. . . . . . . . . . . . . . . . . . . . . . . . . .1

2 文獻回顧. . . . . . . . . . . . . . . . . . . . . . . . . .3
2.1 單維兩母體檢定 . . . . . . . . . . . . . . . . . . . . . . . . . .3
2.2 多維兩母體檢定 . . . . . . . . . . . . . . . . . . . . . . . . . .4
2.3 高維兩母體檢定 . . . . . . . . . . . . . . . . . . . . . . . . . .6
2.3.1 高維兩母體變異矩陣檢定. . . . . . . . . . . . . . . . .7
2.3.2 高維兩母體平均向量檢定. . . . . . . . . . . . . . . . .8

3 研究方法. . . . . . . . . . . . . . . . . . . . . . . . . .12
3.1 PCT-like 統計量與其近似虛無分配 . . . . . . . . . . . . . . . . 12
3.2 E(L) 與 Var(L) 的估計值 . . . . . . . . . . . . . . . . . . . . . 13

4 模擬與實證. . . . . . . . . . . . . . . . . . . . . . . . . .15
4.1 型一誤差率與檢定力模擬 . . . . . . . . . . . . . . . . . . . . . . 15
4.1.1 L(1), L(2)與 PCT 法之模擬比較 . . . . . . . . . . . . . 15
4.1.2 L(1)、 L(2)、 Yao 和 Kim 法之模擬比較 . . . . . . . . . . 16
4.1.3 多維變異矩陣檢定之模擬比較 . . . . . . . . . . . . . . . 16
4.2 實證 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2.1 柏拉圖的著作文集 . . . . . . . . . . . . . . . . . . . . . 17
4.2.2 大腸癌 DNA 微陣列資料 . . . . . . . . . . . . . . . . . 18

5 結論. . . . . . . . . . . . . . . . . . . . . . . . . .20
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Behrens, W. V. (1929), ”Ein Beitrag zur Fehlerberechnung bei wenigen Beobachtungen [A Contribution to Error Estimation with Few Observations]”, Landwirtschaftliche Jahrbücher, 68, 807-837.

Bennett, B. M. (1951), ”Note on a Soulution of the Generalized Behrens-Fisher Problem”, Annals of the Institute of Statistical Mathematics, 2, 87-90.

Box, G. E. P. (1949), ”A General Distribution Theory for a Class of Likelihood Criteria”, Biometrika, 36, 317-346.

Christensen, W. F., and Rencher, A. C. (1997), ”A Comparison of Type I Error Rates and Power Levels for Seven Solutions to the Multivariate Behrens-Fisher Problem”, Communications in Statistics - Simulation and Computation, 26, 1251-1273.

Dempster, A. P. (1958), ”A High Dimensional Two Sample Significance
Test”, The Annals of Mathematical Statistics, 29, 995-1010.

Dempster, A. P. (1960), ”A Significance Test for the Separation of Two Highly Multivariate Small Samples”, Biometrics, 16, 41-50.

Hotelling, H. (1931), ”The Generalization of Student’s Ratio”, The Annals of Mathematical Statistics, 2, 360-378.

James, G. S. (1954), ”Tests of Linear Hypotheses in Univariate and Multivariate Analysis when the Ratios of the Population Variances are Unknown”, Biometrika, 41, 19-43.

Kaluscha, W. (1904), ”Zur Chronologie der platonischen Dialoge”, Wiener Studien, 25-27.

Kim, S. (1992), ”A Practical Solution to the Multivariate Behrens-Fisher Problem”, Biometrika, 79, 171-176.

Kim, S. and Cohen, A. S. (1998), ”On the Behrens-Fisher Problem: A Review”, Journal of Educational and Behavioral Statistics, 23, 356-377.

Krishnamoorthy, K. and Yu, J. (2004), ”Modified Nel and van der Merwe Test for the Multivariate Behrens-Fisher Problem”, Statistics and Probability Letters, 66, 161-169.

Nel, D. G. and van der Merwe, C. A. (1986), ”A Solution to the Multivariate Behrens-Fisher Problem”, Communications in Statistics - Theory and Methods, 15, 3719-3735.

Scheffé, H. (1943), ”On Solutions of the Behrens-Fisher Problem, Based on the t-Distribution”, The Annals of Mathematical Statistics, 14, 35-44.

Schott, J. R. (2007), ”A High-Dimensional Test for the Equality of the Smallest Eigenvalues of a Covariance Matrix”, Journal of Multivariate Analysis, 97, 827-843.

Srivastava, M. S. (2004), ”Multivariate Theory for Analyzing High-Dimensional Data”, Technical Report, University of Toronto, Toronto, Canada.

Srivastava, M. S. (2007), ”Testing the Equality of Two Covariance Matrices and Independence of Two Sub-Vectors with Fewer Observations than the Dimension”,
International Conference on Advances in Interdisci-
plinary Statistics and Combinatorics, Oct 12-14, 2007, University of North Carolina at Greensboro, 2007, NC, USA.

Srivastava, M. S. and Du M. (2008), ”A Test for the Mean Vector with Fewer Observations than the Dimension”, Journal of Multivariate Analysis, 99, 386-402.

Srivastava, M. S. and Yanagihara, H. (2010), ”Testing the Equality of Several Covariance Matrices with Fewer Observations Than the Dimension”, Journal of Multivariate Analysis, 101, 1319-1329.

Welch, B. L. (1938), ”The Significance of the Difference Between Two Means when the Population Variances are Unequal”, Biometrika, 29, 350-362.

Welch, B. L. (1947), ”The Generalization of ‘Student’s’ Problem when Several Different Population Variances are Involved”, Biometrika, 34, 28-35.

Wu, Y., Genton, M. G. and Stefanski, L. A. (2006), ”A Multivariate Two-Sample Mean Test for Small Sample Size and Missing Data”, Biometrics, 62, 877-885.

Yao, Y. (1965), ”An Approximate Degrees of Freedom Solution to the Multivariate Behrens Fisher Problem”, Biometrika, 52, 139-147.

Zhang, J. and Xu, J. (2009), ”On the k-Sample Behrens-Fisher Problem for High-Dimensional Data”, Science in China Series A: Mathematics, 52,1285-1304.
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