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系統識別號 U0002-1401201316141800
中文論文名稱 不等變異數時多種平均數相等檢定方法比較
英文論文名稱 Comparison of several statistics testing the equality of means under heteroscedasticity
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 101
學期 1
出版年 102
研究生中文姓名 吳冠逸
研究生英文姓名 Kuan-Yi Wu
學號 699190327
學位類別 碩士
語文別 中文
口試日期 2012-12-27
論文頁數 62頁
口試委員 指導教授-陳順益
委員-吳秀芬
委員-賴耀宗
中文關鍵字 二階段和一階段樣本  不等變異數  可控制的檢定力  t分配 
英文關鍵字 Two-stage and one-stage sampling  Unequal variances  Controllable power  t-distribution 
學科別分類 學科別自然科學數學
中文摘要 在常態分佈母體下,一般常使用的變異數分析是Fisher的F統計量來檢定各組平均數是否相等。當變異數未知且不相等時,Brown和Forsythe (1974) 針對小樣本的情況下提出Welch,James,ANOVA F* 等單一樣本檢定統計方法來檢定母體平均數是否相等。Bishop和Dudewicz (1978)提出二階段抽樣程序,檢定常態分佈下不等變異數時是否有相同的平均數。而Chen(2001)提出一階段抽樣程序檢定當變異數未知且不相等時是否有相同的平均數。本論文利用電腦模擬計算,以型一誤差和檢定力來比較以上各種檢定統計方法的優劣。
英文摘要 The F-test of equality of normal means in the conventional analysis of variances (ANOVA) is based on the assumption of equal variances. When the variances are unknown and unequal, Brown and Forsythe (1974) compared four single-sample test statistics, Welch, James, ANOVA F and F*, for the equality of normal population means in the case of small sample. Bishop and Dudewicz (1978) proposed a two-stage sampling procedure to test the equality of means under heteroscedasticity. Chen (2001) derived an exact test using one-stage sampling procedure to test the equality of means when the variances are unknown and unequal. In this thesis we employ computer simulation to compare the level and power of the above-mentioned test statistic procedures.
論文目次 目錄
第1章 緒論.....1
第2章 文獻回顧.....3
2.1 二階段抽樣程序 (Two-stage sampling procedure).....3
2.2 一階段抽樣程序 (One-stage sampling procedure).....5
2.3 其他常用的不等變異數分析檢定方法.....6
第3章 單一樣本檢定統計方法的模擬比較.....9
3.1 模擬型一誤差與檢定力.....9
3.2 模擬結果.....11
3.2.1 型一誤差.....11
3.2.2 檢定力.....19
第4章 二階段抽樣程序與單一樣本檢定統計方法的模擬比較.....35
4.1 總樣本數.....35
4.2 模擬結果.....36
4.2.1 檢定力.....36
第5章 結論.....49
參考文獻.....55
附錄 程式碼.....57

表目錄
表1:F、F*、W、J、F ̃^1 之模擬型一誤差I=4,6,10。.....15
表2:F、F*、W、F ̃^1 之模擬檢定力,樣本數為(11,16,16,21)。.....21
表3:F、F*、W、F ̃^1 之模擬檢定力,樣本數為(11,11,16,16,21,21)。.....27
表4:F、F*、W、F ̃^1 之模擬檢定力,樣本數為(20,20,25,25,30,30,35,35,40,40)。.....33
表5:二階段抽樣程序之最後總樣本數。.....39
表6:F*、W、F ̃^1、F ̃^2 之模擬檢定力,變異數為(1,1,1,1)。.....41
表7:F*、W、F ̃^1、F ̃^2 之模擬檢定力,變異數為(1,4,4,9)。.....43
表8:F*、W、F ̃^1、F ̃^2 之模擬檢定力,變異數為(1,1,1,9)。.....45
表9:F*、W、F ̃^1、F ̃^2 之模擬檢定力,變異數為(1,4,9,16)。.....47
表10:各種組合之下的模擬檢定力。.....50

參考文獻 [1]Bartlett, M.S., 1937. Properties of sufficiency and statistical tests. Applied StatisticsJournal of the Royal Statistical Society Series A 160, 268–282.

[2]Bishop,T.A. and Dudewicz,E.J.(1978). Exact analysis of variance with unequal variances: Test Procedures and Tables. Technometrics,20,419-430.

[3]Brown,M.B. and Forsythe,A.B.(1974). The Small Sample Behavior of Some Statistics Which Test the Equality of Several Means. Technometrics,16,129-132.

[4]Chen,S.Y. and Chen,H.J.(1998). Single-stage analysis of variance under heteroscedasicity. Communications in Statistics-simulation and computation,27(3),641-666.

[5]Chen,S.Y.(2001). One-stage and two-stage statistical inference under heteroscedasicity. Communications in Statistics-simulation and computation,30(4),991-1009.

[6]Hartley, H.O. 1950. The Maximum F-ratio as a short-cut test for heterogeneity of variance. Biometrika ,37,308–312.

[7]Levene, H., 1960. In: Olkin, I.I. (Ed.), Contributions to Probability and Statistics:Essays in Honor of Harold Hotelling. Stanford University Press, Palo Alto, CA,pp. 278–292.

[8]R.G.Krutchkoff.(1988). One way fixed effects analysis of variance when the error varianes may be unequal. Journal of Statistical Computation and Simulation,30,259-271

[9]Samaradasa,Weerahandi.(1995). ANOVA under Unequal Error Variances. Biometrics,51,589-599.

[10]Stein,C.M.(1945). A two-sample test for a linear hypothesis whose power is independent of the variance. Annals of Mathematical Statistics,16,243-258.
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