不等變異數時多種平均數相等檢定方法比較

Comparison of several statistics testing the equality of means under heteroscedasticity

吳冠逸(Kuan-Yi Wu)

指導教授 ： 陳順益

淡江大學/理學院/數學學系碩士班/碩士(2013年)

https://doi.org/10.6846/TKU.2013.00366

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。

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

並列摘要

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.

並列關鍵字

Two-stage and one-stage sampling ； Unequal variances ； Controllable power ； t-distribution

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.