臺灣博碩士論文加值系統

常態分配時常會被拿來當作許多分配生成的基底，也有許多分配就是利用常態分配所衍生而來的。所以當我們在檢定資料是否來自常態分配是很重要的一個環節，然而我們現行所熟知檢定常態分配的方法不外乎有Normal probability QQ-plot、Kolmogorov-Smirnov Test、以及Shapiro-Wilk test，但每種方法都有其優點和其比較不適用或是缺失的狀況下。例如Normal probability QQ-plot沒有數學的檢定公式只光靠其圖形的判斷但每個人的圖形主觀意識不同而有不同的結果，Kolmogorov-Smirnov Test是利用近似分配來估計所以不免會有誤差而影響到結果的判斷。Shapiro-Wilk test則因為其檢定統計量的公式有較複雜的數學運算尤其是在樣本數較大的時候更顯困難。然而最重要的是上述所提到的三種方法都是在單一維度狀況下才適用，但我們處理的資料或許不再只是只有單一維度這樣簡單了所以要有一個能適用在多維度狀況下的方法是必要的，有鑑於此我們研究利用漸進卡方分配的方法並利用相關係數公式來當作檢定統計量，並且在後面會介紹其檢定力大小的計算來看每個不同的對立假設下其判斷。而利用我們研究的方法最後面可看出在樣本數較大的時候其檢定統計量會大於我們所模擬的查表值，因此不拒絕虛無假設也就是說資料符合我們預期的常態分配。所以我們所提供的方法不但可適用於單一維度更可推廣至多維度，前面我們介紹的3種方法都是相當好的檢定，然而我們的研究更提供了多維度的條件下以供參考。

Normal distribution usually is taken to be the basic of distribution generation. Also, there are several distributions deriving by normal distribution. Therefore, it’s an important link when we are testing the data comes from normal distribution or not. However, the existent methods of testing normal distribution we are familiar with are Normal probability, QQ-plot, kolmogorov-smirnov test, and Shapiro-Wilk test. But every method has its own advantage and inappropriate or defect situations. For example, Normal probability QQ-plot does not have mathematical testing formula, and it is merely determined by graphs. As the result which depends on the personal subjective sense has the different conclusion. Kolmogorov-smirnov test uses approximate distribution to estimate, but the conclusion will be affected by bias. As the Shapiro-Wilk test, the formula of test statistic has more complicated operation especially obvious difficulty with huger sample size. However the most important thing is that three methods mentioned above are applicative under univariate dimension. The data we arrange would not be so simple like univariate dimension anymore. It is necessary to find a method that adapt to multivariate dimension. We use method of asymptotic chi-square distribution and formula of correlation coefficient to be test statistic. I would like to introduce estimation of individual alternative hypothesis by the calculation of power. We can find that the test statistic would greater than the critical points we approximated while the sample size is bigger. Consequently, we do not reject Null hypothesis, and it means the data and normal distribution we anticipated in accordance. Therefore, the methods we applied not only can suit to univariate dimension, but also popularize in multivariate dimension. These three methods I introduced are quite excellent testing. However our research further provide more reference to fit in multivariate dimension.

摘要
Abstract
目錄
圖目錄
表目錄
1序論
2二維常態分配
2.1收斂分配
2.2 二維常態分配下之Q-Q plot
2.3研究方法
3模擬
3.1二維常態分配
3.2三維常態分配
4例子
4.1檢定力
4.2實際例子
5結論
參考文獻
圖目錄
圖2.1 樣本數為10的QQ plot
圖2.2 樣本數為100的QQ plot
表目錄
表3-1 二維常態分配模擬表
表3-2 三維常態分配模擬表
表4-1 檢定力表
表4-2 檢定力表
表4-3 檢定力表
表4-4 檢定力表
表4-5 三種品種蝴蝶花
表4-6 各個品種長寬之檢定統計量
表4-7 蕚片及花瓣長寬檢定統計量

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[2] Fisher,R.A.(1936) The use of multiple measurement in taxonomic problems Ann.Eugen. Lond,8,376.
[3] Johnson , Richard 1998, Applied Multivariate Statistical Analysis
[4] Kolomogorov , A.N (1933) Sulla determinazione empirica di una legge di distribution. Giornal dell’s Istituto Italiano degli Attuari,4,89-91(6.1)
[5] S.James prese, “Multivariate analysis” (1971)