在檢定母體平均數是否相等的變異數分析方法中，其基本假設是模型誤差項為獨立的常態分布、和有相等變異數。當變異數未知且不相等時，Bishop and Dudewicz (1978) 導出二階段抽樣程序方法，而 Chen (2001) 提出一階段抽樣程序方法，分別用來檢定是否有相同的母體平均數。本文首先討論一階段與二階段抽樣程序在同樣的總樣本數下，比較不同的起始樣本n0 的表現。此外在模型誤差項為不同分布時，討論兩種抽樣程序的型一誤差與檢定力表現。

One of the assumptions of test procedures in the conventional analysis of variance is the equality of error variances. When the variances are unknown and unequal, Bishop and Dudewicz (1978) developed an exact analysis of variance for the means of  k independent normal populations by using a two-stage sampling procedure. Chen and Chen (1998) used a one-stage sampling procedure to test hypotheses of equality of means in ANOVA model. In this study, we will first thoroughly explore the optimal choice of the initial sample size, n_0, for both the one-stage and two-stage sampling procedures by simulation study. We also investigate the effect on inference about the means of the one-stage and two-stage procedures when the assumption of normality is violated.

目錄
第1 章前言············································1
第2 章文獻回顧································........2
2.1 二階段抽樣程序(Two-stage sampling procedure)······2
2.2 一階段抽樣程序(One-stage sampling procedure)······3
第3 章常態分布時一階段抽樣程序與二階段抽樣程序········6
3.1 總樣本數··········································6
3.2 模擬型一誤差與檢定力······························7
3.3 模擬結果··········································7
3.3.1 型一誤差········································8
3.3.2 檢定力··········································9
第4 章常態分布時不同n10 的一階段抽樣程序·············24
4.1 n10 的選取·······································24
4.2 模擬結果·········································25
4.2.1 型一誤差·······································25
4.2.2 檢定力·········································28
第5 章不同分布時一階段抽樣程序與二階段抽樣程序·······43
5.1 三種分布的差異性·································44
5.2 模擬三種分布·····································44
5.3 模擬結果·········································45
5.3.1 均勻分布型一誤差·······························46
5.3.2 均勻分布檢定力·································47
5.3.3 伽瑪分布型一誤差·······························48
5.3.4 伽瑪分布檢定力·································49
5.3.5 對數常態分布型一誤差···························49
5.3.6 對數常態分布檢定力·····························50
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5.4 總結·············································51
參考文獻·············································70
附錄程式·············································71

表目錄
3.1 樣本數U,L,M之值, 變異數為(1, 1, 1, 1)。···············12
3.2 樣本數U,L,M之值, 變異數為(0.1, 0.4, 0.4, 0.9)。·······13
3.3 樣本數U,L,M之值, 變異數為(1, 4, 4, 9)。···············14
3.4 樣本數U,L,M之值, 變異數為(1, 1, 1, 9)。···············15
3.5 ˜ F1與˜ F2之型一誤差, 變異數為(1, 1, 1, 1)。··········16
3.6 ˜ F1與˜ F2之型一誤差, 變異數為(0.1, 0.4, 0.4, 0.9)。··17
3.7 ˜ F1與˜ F2之型一誤差, 變異數為(1, 4, 4, 9)。··········18
3.8 ˜ F1與˜ F2之型一誤差, 變異數為(1, 1, 1, 9)。··········19
3.9 ˜ F1與˜ F2之檢定力, 變異數為(1, 1, 1, 1)。············20
3.10 ˜ F1與˜ F2之檢定力, 變異數為(0.1, 0.4, 0.4, 0.9)。···21
3.11 ˜ F1與˜ F2之檢定力, 變異數為(1, 4, 4, 9)。···········22
3.12 ˜ F1與˜ F2之檢定力, 變異數為(1, 1, 1, 9)。···········23
4.1 不同n10時˜ F1
U之型一誤差, 變異數為(1, 1, 1, 1)。·······················31
4.2 不同n10時˜ F1
U之型一誤差, 變異數為(0.1, 0.4, 0.4, 0.9)。···············32
4.3 不同n10時˜ F1
U之型一誤差, 變異數為(1, 4, 4, 9)。·······················33
4.4 不同n10時˜ F1
M之型一誤差, 變異數為(1, 1, 1, 1)。·······················34
4.5 不同n10時˜ F1
M之型一誤差, 變異數為(0.1, 0.4, 0.4, 0.9)。···············35
4.6 不同n10時˜ F1
M之型一誤差, 變異數為(1, 4, 4, 9)。·······················36
4.7 不同n10時˜ F1
U之檢定力, 變異數為(1, 1, 1, 1)。·························37
4.8 不同n10時˜ F1
U之檢定力, 變異數為(0.1, 0.4, 0.4, 0.9)。·················38
4.9 不同n10時˜ F1
U之檢定力, 變異數為(1, 4, 4, 9)。·························39
4.10 不同n10時˜ F1
M之檢定力, 變異數為(1, 1, 1, 1)。·························40
4.11 不同n10時˜ F1
M之檢定力, 變異數為(0.1, 0.4, 0.4, 0.9)。·················41
4.12 不同n10時˜ F1
M之檢定力, 變異數為(1, 4, 4, 9)。·························42
ii
5.1 均勻分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 1, 1, 1)。....................................52
5.2 均勻分布時˜ F1與˜ F2之型一誤差, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................53
5.3 均勻分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 4, 4, 9)。....................................54
5.4 均勻分布時˜ F1與˜ F2之之檢定力, 
變異數為(1, 1, 1, 1)。....................................55
5.5 均勻分布時˜ F1與˜ F2之之檢定力, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................56
5.6 均勻分布時˜ F1與˜ F2之之檢定力, 
變異數為(1, 4, 4, 9)。....................................57
5.7 伽瑪分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 1, 1, 1)。....................................58
5.8 伽瑪分布時˜ F1與˜ F2之型一誤差, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................59
5.9 伽瑪分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 4, 4, 9)。....................................60
5.10 伽瑪分布時˜ F1與˜ F2之檢定力, 
變異數為(1, 1, 1, 1)。....................................61
5.11 伽瑪分布時˜ F1與˜ F2之檢定力, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................62
5.12 伽瑪分布時˜ F1與˜ F2之檢定力, 
變異數為(1, 4, 4, 9)。....................................63
5.13 對數常態分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 1, 1, 1)。....................................64
5.14 對數常態分布時˜ F1與˜ F2之型一誤差, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................65
5.15 對數常態分布時˜ F1與˜ F2之型一誤差, 
變異數為(1, 4, 4, 9)。....................................66
5.16 對數常態分布時˜ F1與˜ F2之檢定力, 
變異數為(1, 1, 1, 1)。....................................67
5.17 對數常態分布時˜ F1與˜ F2之檢定力, 
變異數為(0.1, 0.4, 0.4, 0.9)。............................68
5.18 對數常態分布時˜ F1與˜ F2之檢定力, 
變異數為(1, 4, 4, 9)。....................................69
圖目錄
5.1 四種分布在平均數為4 與變異數為9 的機率密度函數........45
iii

Bishop, T.A., Dudewicz, E.J.(1978). Exact analysis of variance with unequal
variances: Test procedures and tanles. Tschnometrics, 20,419-430.
Bradley, J.V.(1978). Robustness? British Journal of Mathematical and Statistical
Psychology, 31, 144-152.
Chen, H.J., Lam, K.(1989). Single-stage interval estimation of the largest normal
mean under heteroscedasicity. Communications in Statistics Theory and Methods,
18(10), 3703-3718.
Chen, S.Y., Chen, H.J.(1998). Single-stage analysis of variance under heteroscedasticity.
Communications in Statistics.-simulation and computation, 27(3), 641-666.
Chen, S.Y. (2001). One-stage and two-stage statistical inference under heteroscedasticity.
Communications in Statistics.-simulation and computation, 30(4), 991-1009.
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.
Wilcox, R.R.(1983) A table of percentage points of the range of independent t
variables. Technometrics, 25(2), 201-204.