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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"><!--Converted with LaTeX2HTML 2002-2 (1.70)original version by: Nikos Drakos, CBLU, University of Leeds* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan* with significant contributions from: Jens Lippmann, Marek Rouchal, Martin Wilck and others --><HTML><HEAD><TITLE>The questions addressed</TITLE><META NAME="description" CONTENT="The questions addressed"><META NAME="keywords" CONTENT="web1"><META NAME="resource-type" CONTENT="document"><META NAME="distribution" CONTENT="global"><META NAME="Generator" CONTENT="LaTeX2HTML v2002-2"><META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css"><LINK REL="STYLESHEET" HREF="web1.css"><LINK REL="next" HREF="node10.html"><LINK REL="previous" HREF="node8.html"><LINK REL="up" HREF="node6.html"><LINK REL="next" HREF="node10.html"></HEAD><BODY ><!--Navigation Panel--><A NAME="tex2html276" HREF="node10.html"><IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/next.png"></A> <A NAME="tex2html274" HREF="node6.html"><IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/up.png"></A> <A NAME="tex2html268" HREF="node8.html"><IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/prev.png"></A> <BR><B> Next:</B> <A NAME="tex2html277" HREF="node10.html">The bootstrap: Some Examples</A><B> Up:</B> <A NAME="tex2html275" HREF="node6.html">Lectures</A><B> Previous:</B> <A NAME="tex2html269" HREF="node8.html">The underlying principle</A><BR><BR><!--End of Navigation Panel--><H1><A NAME="SECTION00230000000000000000">The questions addressed</A></H1>Suppose we are interested in the estimation of an unknownparameter <IMG WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0" SRC="img10.png" ALT="$\theta$">(this doesn't mean that we are in a parametric context).The two main questions asked at that stage are :<BR><UL><LI><B><FONT SIZE="+1">Question 1</FONT></B> What estimator <IMG WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0" SRC="img10.png" ALT="$\theta$"> should be used ?</LI><LI><B><FONT SIZE="+1">Question 2</FONT></B> Having chosen an estimator, how accurate is it ?</LI></UL><P>The second question has to be answered throughinformation on the distribution or at least the variance of the estimator.<P>Of course there are answers in very simple contexts:for instance when the parameter of interest isthe mean <IMG WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0" SRC="img11.png" ALT="$\mu$"> then the estimator<IMG WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="img12.png" ALT="$\bar{X}$"> has a known standarddeviation : the estimated standard error noted sometimes <BR><P></P><DIV ALIGN="CENTER"><!-- MATH \begin{displaymath}\hat{\sigma}=\left[(X_i-\bar{X})^2/n^2\right]\end{displaymath} --><IMG WIDTH="158" HEIGHT="38" BORDER="0" SRC="img13.png" ALT="\begin{displaymath}\hat{\sigma}=\left[(X_i-\bar{X})^2/n^2\right]\end{displaymath}"></DIV><BR CLEAR="ALL"><P></P>However no such estimator is available for the sample median for instance.<P>In maximum likelihood theory the question1 is answered through using the mle and then thequestion 2 can be answered with an approximatestandard error of <!-- MATH $\widehat{se}(\theta)=\frac{1}{\sqrt{Fisher Info}}$ --><IMG WIDTH="162" HEIGHT="41" ALIGN="MIDDLE" BORDER="0" SRC="img14.png" ALT="$\widehat{se}(\theta)=\frac{1}{\sqrt{Fisher Info}}$"><P>The bootstrap is a more general way to answerquestion 2 , with the following aspects:<UL><LI>Less or no parametric modelling.</LI><LI>More computation. (a factor 100 to 1000)</LI><LI>Automatic, whatever the situation (can be complex).</LI></UL><P>If we had several samples from the unknown (true) distribution<IMG WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="img1.png" ALT="$F$"> then we could consider the variationsof the estimator :<BR><P></P><DIV ALIGN="CENTER"><!-- MATH \begin{displaymath}\begin{array}{llllll}F & \stackrel{\mbox{Random Sample}}{\longrightarrow} &(X_1^1,X_2^1...X_n^1) & = & {\cal X}_n^1 & \hat{\theta}_1\\& \vdots & \vdots & \vdots & \vdots\\& \stackrel{\mbox{Random Sample}}{\longrightarrow} &(X_1^2,X_2^2...X_n^2) & = & {\cal X}_n^2 & \hat{\theta}_2\\& \vdots & \vdots & \vdots & \vdots\\& \stackrel{\mbox{Random Sample}}{\longrightarrow} &(X_1^B,X_2^B...X_n^B) & = & {\cal X}_n^B & \hat{\theta}_B\end{array}\end{displaymath} --><IMG WIDTH="413" HEIGHT="172" BORDER="0" SRC="img15.png" ALT="\begin{displaymath}\begin{array}{llllll}F & \stackrel{\mbox{Random Sample}}{\l......,X_2^B...X_n^B) & = & {\cal X}_n^B & \hat{\theta}_B\end{array}\end{displaymath}"></DIV><BR CLEAR="ALL"><P></P>Such a situation is never the case, so we replace thesenew samples by a resampling procedure based onthe only information we have about <IMG WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0" SRC="img1.png" ALT="$F$">,and that is an empirical <IMG WIDTH="26" HEIGHT="45" ALIGN="MIDDLE" BORDER="0" SRC="img16.png" ALT="$\hat{F}_n$"> :thisside is what is called <B>bootstrapping</B>.<P><HR><!--Navigation Panel--><A NAME="tex2html276" HREF="node10.html"><IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/next.png"></A> <A NAME="tex2html274" HREF="node6.html"><IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/up.png"></A> <A NAME="tex2html268" HREF="node8.html"><IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous" SRC="file:/home/depot/swtree/depot/latex2html-2002-2/latex2html-2002-2/icons/prev.png"></A> <BR><B> Next:</B> <A NAME="tex2html277" HREF="node10.html">The bootstrap: Some Examples</A><B> Up:</B> <A NAME="tex2html275" HREF="node6.html">Lectures</A><B> Previous:</B> <A NAME="tex2html269" HREF="node8.html">The underlying principle</A><!--End of Navigation Panel--><ADDRESS>Susan Holmes2004-05-19</ADDRESS></BODY></HTML>⌨️ 快捷键说明
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