Normal Approximation for Random Vectors

• Main Authors: Ting-Ting Chuang, 莊婷婷
• Other Authors: Yuh-Jia Lee
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/14693461166527256746

碩士 === 國立高雄大學 === 統計學研究所 === 97 === Unlike the Stein's method introduced in his celebrated paper\cite{S}, we consider the following alternative Stein's equation \begin{equation}\label{se1a} F''(w)-wF'(w)=\tilde{h}, \end{equation} where $\tilde{h}:=-\mathbb{E}h(Z)$ and $Z$ is a standard normal distributed random variable. The corresponding Stein identity now becomes \[ \mathbb{E}[F''(W)]=\mathbb{E}[WF'(W)], \] which also characterized the standard normal random variable as well. The solution of (\ref{se1a}) $F=F_{\tilde{h}}$ 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}dy dt. \end{eqnarray} The advantage of adapting equation (\ref{se1a}) is not only that the solution (\ref{sola}) is itself very easy to handle but also that the solution is ready to be extended to vector-valued random variable. 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 \[ F_{\tilde{h}}(w)=\int_{0}^{\infty}\left(\frac{1}{\sqrt{2\pi}}\right)^n\int_{\mathbb{R}^n}\tilde{h}(e^{-t}w+\sqrt{1-e^{-2t}}y)e^{-|y|^2/2}dy dt. \] In this paper we reprove the key lemma (Lemma 2.3 in \cite{BC}) and then obtain the estimation of upper bound for Wasserstein distance and Kolmogorov distance.\\ Keyword: Stein's method, Stein's equation, Stein identity, random vectors, Wasserstein distance, Kolmogorov distance.