將偏態列入考慮的中央極限定理應用

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Title:	將偏態列入考慮的中央極限定理應用
Other Titles:	Taking skewness into consideration when applying the central limit theorem
Authors:	翁新傑;Wong, Hsin-Chieh
Contributors:	淡江大學數學學系碩士班鄭惟厚;Cheng, Wei-Hou
Keywords:	中央極限定理;檢定;偏態;Edgeworth展開式;Central limit theorem;hypotheses testing;Skewness;Edgeworth expansion
Date:	2012
Issue Date:	2013-04-13 11:12:08 (UTC+8)
Abstract:	在應用中央極限定理做統計推論時，樣本大小n必須夠大，但多大才足夠則並無定論，和母體分布有關，其中一個重要的影響因素是分布的偏斜程度。偏斜程度較大的分布，樣本大小n需要更大才會接近中央極限定理的結果。當我們檢定H_0:μ=μ_0對應H_1:μ>μ_0時，根據許力升論文的討論，以樣本標準差S_n取代母體標準差σ時P(Type I error)低於預期之0.05，而我們想到若在應用中央極限定理做檢定時，將偏斜程度列入考慮做些調整，結果可能會比起直接應用要來得好。分布的偏斜程度可以使用偏態(skewness)來表達，因為Edgeworth expansion當中有一項係數包含了偏態在內，所以我們將利用它來探討這個問題，並將建議如何利用Edgeworth expansion對檢定問題的臨界值做修正。我們以電腦模擬方式，以λ家族所產生的四個分布進行討論，在σ已知的條件下討論結果發現，比起只用傳統的中央極限定理來做檢定，利用Edgeworth expansion所模擬得出的α值幾乎都比較接近原先所設定的α值，在偏態較大，而n較小時，差別更為明顯。在σ未知的條件下討論結果發現，雖然σ和偏態都是用估計值代替，但除了樣本大小n為5與10的情況下，比起只用傳統的中央極限定理來做檢定，利用Edgeworth expansion所模擬得出的α值幾乎都比較接近原先所設定的α值，符合我們的預期。 When applying the central limit theorem on statistical inference, sample size n has to be large, but different text books give different suggestions on how large the sample size should be. We observed that the skewness of the population plays an important role in this matter. When the distribution of the population is very skewed, it takes a bigger sample size for the distribution of the sample mean X.bar to get close to the normal distribution. In this paper we are interested in the problem of testing H_0:μ=μ_0vs.H_1:μ>μ_0. In Li-Sheng Hsu’s master’s thesis he noted that when the population standard deviation is unknown and has to be replaced by the sample standard deviation, the probability of a Type I error is often a lot smaller than the designated α of 0.05. In this paper we want to take skewness into consideration and try to cut down the difference between the actual and designated significant levels. Edgeworth expansion was used and we were successful in making adjustments to the critical value to achieve our goal, shown by the results of computer simulations.
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