國立高雄大學圖資館 |

Language: English

Back

隨機向量之常態逼近 = Normal Approximation for...

國立高雄大學統計學研究所

隨機向量之常態逼近 = Normal Approximation for Random Vectors

Record Type:	Language materials, printed : monographic
Paralel Title:	Normal Approximation for Random Vectors
Author:	莊婷婷,
Secondary Intellectual Responsibility:	國立高雄大學
Place of Publication:	[高雄市]
Published:	撰者;
Year of Publication:	2009[民98]
Description:	21面圖、表 : 30公分;
Subject:	Kolmogorov 測距
Subject:	Kolmogorov distance.
Online resource:	http://handle.ncl.edu.tw/11296/ndltd/07891702086273512503
Notes:	指導教授：李育嘉
Notes:	參考書目：面
Summary:	不同於在Stein著名論文\cite{S}中所提到的Stein方法，我們可考慮Steinequation的另一種形式\begin{equation}F''(w)-wF'(w)=\tilde{h},\end{equation}其中 $\tilde{h}:=h-\mathbb{E}h(Z)$且 $Z$為一隨機變數服從標準常態分佈。其對應的Stein identity便可表示成\[\mathbb{E}[F''(W)]=\mathbb{E}[WF'(W)],\]也可利用此等式來刻劃出標準常態分佈的隨機變數。而上述式子的解，我們也可將其表示為\begin{eqnarray}F_{\tilde{h}}(w)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-y^2/2}dydt.\end{eqnarray}這樣表示法的優點不但可簡化求解的過程，而且也可將其推廣至n維的情況，這是原來方法所無法處理的部分。舉例來說，n維的隨機向量其Steinequation可寫成\[\Delta F(w)-w\cdot \nabla F(w)=\tilde{h},\]而Stein identity可寫成\[\mathbb{E}[\Delta F(W)]=\mathbb{E}[W\cdot \nabla F(W)]\]那麼，我們將便可將其解表示成\[F_{\tilde{h}}(w)=\int_{0}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^n\int_{\mathbb{R}^n}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-\|y\|^2/2}dydt.\]在這篇論文中，我們將重新證明其主要引理(參見\cite{BC}，引理2.3)，並得到Wassersteindistance和Kolmogorov distance的上界估計。 Unlike the Stein's method introduced in his celebratedpaper\cite{S}, we consider the following alternative Stein'sequation\begin{equation}\label{se1a}F''(w)-wF'(w)=\tilde{h},\end{equation}where\tilde{h}:=-\mathbb{E}h(Z)and Z is a standard normaldistributed random variable. The corresponding Stein identity nowbecomes\[\mathbb{E}[F''(W)]=\mathbb{E}[WF'(W)],\]which also characterized the standard normal random variable aswell. The solution of (\ref{se1a}) is given by\begin{eqnarray}\label{sola}F_{\tilde{h}}(w)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-y^2/2}dydt.\end{eqnarray}The advantage of adapting equation (\ref{se1a}) is not only that thesolution (\ref{sola}) is itself very easy to handle but also thatthe solution is ready to be extended to vector-valued randomvariable. For example, for random vectors taking values in\mathbb{R}^n\|, the Stein equation (\ref{se1a}) becomes\[\Delta F(w)-w\cdot \nabla F(w)=\tilde{h},\]the Stein identity becomes\[\mathbb{E}[\Delta F(W)]=\mathbb{E}[W\cdot \nabla F(W)],\] and the solution (\ref{sola}) now becomes\[F_{\tilde{h}}(w)=\int_{0}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^n\int_{\mathbb{R}^n}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-\|y\|^2/2}dydt.\] In this paper we reprove the key lemma(Lemma 2.3 in \cite{BC}) and then obtain the estimation of upperbound for Wasserstein distance and Kolmogorov distance.

隨機向量之常態逼近 = Normal Approximation for Random Vectors
莊, 婷婷

隨機向量之常態逼近 = Normal Approximation for Random Vectors / 莊婷婷撰 - [高雄市] : 撰者, 2009[民98]. - 21面 ; 圖、表 ; 30公分.
指導教授：李育嘉參考書目：面.
Kolmogorov 測距Kolmogorov distance.

隨機向量之常態逼近 = Normal Approximation for Random Vectors
LDR:03789nam0a2200277 450 001 220252
005 20170214100909.0
009 220252
010 0 $b 精裝
010 0 $b 平裝
100 $a 20170214y2009 k y0chiy09 ea
101 1 $a eng $d chi $d eng
102 $a tw
105 $a ak am 000yy
200 1 $a 隨機向量之常態逼近 $d Normal Approximation for Random Vectors $z eng $f 莊婷婷撰
210 $a [高雄市] $c 撰者 $d 2009[民98]
215 0 $a 21面 $c 圖、表 $d 30公分
300 $a 指導教授：李育嘉
300 $a 參考書目：面
328 $a 碩士論文--國立高雄大學統計學研究所
330 $a 不同於在Stein著名論文\cite{S}中所提到的Stein方法，我們可考慮Steinequation的另一種形式\begin{equation*}F''(w)-wF'(w)=\tilde{h},\end{equation*}其中 $\tilde{h}:=h-\mathbb{E}h(Z)$且 $Z$為一隨機變數服從標準常態分佈。其對應的Stein identity便可表示成\[\mathbb{E}[F''(W)]=\mathbb{E}[WF'(W)],\]也可利用此等式來刻劃出標準常態分佈的隨機變數。而上述式子的解，我們也可將其表示為\begin{eqnarray*}F_{\tilde{h}}(w)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-y^2/2}dydt.\end{eqnarray*}這樣表示法的優點不但可簡化求解的過程，而且也可將其推廣至n維的情況，這是原來方法所無法處理的部分。舉例來說，n維的隨機向量其Steinequation可寫成\[\Delta F(w)-w\cdot \nabla F(w)=\tilde{h},\]而Stein identity可寫成\[\mathbb{E}[\Delta F(W)]=\mathbb{E}[W\cdot \nabla F(W)]\]那麼，我們將便可將其解表示成\[F_{\tilde{h}}(w)=\int_{0}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^n\int_{\mathbb{R}^n}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-|y|^2/2}dydt.\]在這篇論文中，我們將重新證明其主要引理(參見\cite{BC}，引理2.3)，並得到Wassersteindistance和Kolmogorov distance的上界估計。 Unlike the Stein's method introduced in his celebratedpaper\cite{S}, we consider the following alternative Stein'sequation\begin{equation}\label{se1a}F''(w)-wF'(w)=\tilde{h},\end{equation}where\tilde{h}:=-\mathbb{E}h(Z)and Z is a standard normaldistributed random variable. The corresponding Stein identity nowbecomes\[\mathbb{E}[F''(W)]=\mathbb{E}[WF'(W)],\]which also characterized the standard normal random variable aswell. The solution of (\ref{se1a}) is given by\begin{eqnarray}\label{sola}F_{\tilde{h}}(w)=\int_{0}^{\infty}\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-y^2/2}dydt.\end{eqnarray}The advantage of adapting equation (\ref{se1a}) is not only that thesolution (\ref{sola}) is itself very easy to handle but also thatthe solution is ready to be extended to vector-valued randomvariable. For example, for random vectors taking values in\mathbb{R}^n|, the Stein equation (\ref{se1a}) becomes\[\Delta F(w)-w\cdot \nabla F(w)=\tilde{h},\]the Stein identity becomes\[\mathbb{E}[\Delta F(W)]=\mathbb{E}[W\cdot \nabla F(W)],\] and the solution (\ref{sola}) now becomes\[F_{\tilde{h}}(w)=\int_{0}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^n\int_{\mathbb{R}^n}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-|y|^2/2}dydt.\] In this paper we reprove the key lemma(Lemma 2.3 in \cite{BC}) and then obtain the estimation of upperbound for Wasserstein distance and Kolmogorov distance.
510 1 $a Normal Approximation for Random Vectors $z eng
610 0 $a Kolmogorov 測距 $a Stein 方程 $a Stein 等式 $a Stein方法 $a Wasserstein 測距 $a 隨機向量
610 1 $a Kolmogorov distance. $a Stein identity $a Stein's equation $a Stein's method $a Wasserstein distance $a random vectors
681 $a 008M/0019 $b 343201 4444 $v 2007年版
700 1 $a 莊 $b 婷婷 $4 撰 $3 353927
712 0 2 $a 國立高雄大學 $b 統計學研究所 $3 166081
801 0 $a tw $b 國立高雄大學 $c 20091020 $g CCR
856 7 $2 http $u http://handle.ncl.edu.tw/11296/ndltd/07891702086273512503