臺灣博碩士論文加值系統

雷諾應力為紊流流場中帶動泥沙運動的主要動力之一，而雷諾應力又可表示為二隨機變數之乘積，本研究探討變數乘積機率值的計算方法，以估算雷諾應力。依據Wood, Booth &; Butler(1993)所提出推廣典型鞍點近似法理論，本研究推導了以雙指數分佈為基底之鞍點近似法來概算變數乘積的尾部機率值，稱為改良鞍點近似法，並與常態近似法及典型的鞍點近似法作比較。

在評估過程中考慮四種情況：
a 兩變數分別為常態分佈，且兩變數為獨立；
b 兩變數分別為常態分佈，且兩變數為相依；
c 一變數為常態分佈，另一變數為合成常態分佈，且兩變數為獨立；
d 一變數為常態分佈，另一變數為合成常態分佈，且兩變數為相依；

由研究結果得知，在兩常態變數乘積情況下，典型與改良的鞍點近似法皆表現比常態近似法精準；而在一常態與一合成常態的情形中，改良的鞍點近似法比另兩種近似法來的準確，當合成常態變數的峰態係數較大時，改良的鞍點近似法有更明顯的近似效果。

Reynolds stress, which can be expressed as a product of two random variables, is one of the main driving forces to initiate the sediment movement for turbulence flow. In order to estimate the Reynolds stress, the method to calculate the tail probability of the product of two random variables was investigated in this paper.

Based on Wood,Booth&Butler(1993) generalization of the typical saddlepoint approximation theorem, a saddlepoint approximation with the double-expontial base was proposed to approximate the tail probability of the product of two random variables. The proposed method was named modified saddlepoint appromimation in this study. Furthermore, this new method was compared with the normal approximation and the typical saddlepoint approximation.

The following four different cases were considered:
a) the product of two independent normal variables
b) the product of two dependent normal variables
c) the product of a normal variable and a contaminated normal variable, which
are independent
d) the product of a normal variable and a contaminated normal variable, which
are dependent

According to the results of this study, the typical and the modified saddlepoint approximations are more accurate than the normal approximation under the cases of (a) and (b). For the cases of (c) and (d), the modified saddlepoint approximation is more accurate than the other two approximations, which was especially true when the kurtosis of the contaminated normal variable was large.

Chapter 1緒論
1.1研究背景
1.2研究目的
1.3研究內容
Chapter 2文獻回顧
2.1鞍點近似法的起源
2.2尾部機率的鞍點近似法
2.3 非常態為基底的鞍點近似法
2.4 利用鞍點近似法概算兩乘積分布
Chapter 3 變數乘積之鞍點近似法
3.1 鞍點近似法之步驟
3.1.1 典型的鞍點近似法
3.1.2 改良的鞍點近似法
3.2 兩變數之乘積
3.2.1 兩獨立常態乘積分佈
3.2.2 兩相依常態乘積分佈
3.2.3 合成變數的尾部機率值
3.2.3 獨立的常態與合成常態乘積之分佈
3.2.4 相依的常態與合成常態乘積之分佈
Chapter 4 研究結果
4.1 兩獨立常態之近似法比較
4.2兩相依常態之近似法比較
4.3 獨立的常態與合成常態之近似法比較
4.4 相依的常態與合成常態之近似法比較
4.5 真實資料的估計
Chapter 5 結論及建議
5.1 結論
5.2 建議
附錄
A 合成常態變數是一長尾資料之證明
B 圖形
B.1 兩獨立常態乘積
B.2 兩相依常態乘積
B.3 獨立的常態與合成常態乘積
B.4 相依的常態與合成常態
C 參考文獻

A. T. A. Wood, J. G. Booth, and R. W. Butler (1993), "Saddlepoint
approximations to the CDF of some statistics with nonnormal limit
distributions" J. Amer. Statist. Assoc. 88 680-686

A. T. A. Wood, J. G. Booth,(1994) "An example in which the Lugannani-Rice
saddlepoint formula fails" Stat. Prob. Letters 23 53-61

C. Goutis, , and G. Casella (1999), "Explaining the Saddlepoint approximation"
The American Statistician, 53:216-224

J. L. Devore(2004), "Probability and statistics", sixth edition,Duxbury.

G. G. Roussas(1997), "A course in mathematical statistics", second edition,
Academic Press. G. Casella and R. L.Berger(2002), "Statistical
Inference "second edition, Duxbury.

H. E. Daniels, (1987), "Tail Probability Approximations," International
Statistical Review, 55, 37-48

N. L. Johnson and S. kotz(1970), "Continuous univariate distribution1",Wiley.

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陳秋帆(2002)，以鞍點近似法概算變數乘積之分配，靜宜大學應用數學系碩士論文。

彭國倫 編著(1997)，Fortran 90程式設計，碁峰資訊。

張智星 著(2000)，Matalb 程式設計與應用，清蔚科技。