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📁 matlab bootstrap程序设计方法
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<H2><A name=SECTION00311000000000000000>In practice</A> </H2>From an original 
sample <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}{\cal X}_n=(X_1,X_2...X_n) \stackrel{iid}{\sim} F\end{displaymath} --><IMG 
height=39 
alt="\begin{displaymath}{\cal X}_n&#10;=(X_1,X_2...X_n) \stackrel{iid}{\sim} F\end{displaymath}" 
src="In practice.files/img290.png" width=29 border=0> </DIV><BR clear=all>
<P></P>draw a new sample of <IMG height=18 alt=$n$ 
src="In practice.files/img10.png" width=16 align=bottom border=0> observations 
among the original sample with replacement, each observation having the same 
probabilty of being drawn (<IMG height=32 alt=$=\frac{1}{n}$ 
src="In practice.files/img291.png" width=486 align=middle border=0>). The 
bootstrap sample is often denoted <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}{\cal X}_n^*=X_1^*,X_2^*...X_n^* \stackrel{iid}{\sim} F_n\mbox{ the empirical distribution }\end{displaymath} --><IMG 
height=46 
alt="\begin{displaymath}{\cal X}_n^*&#10;=X_1^*,X_2^*...X_n^* \stackrel{iid}{\sim} F_n&#10;\mbox{ the empirical distribution }\end{displaymath}" 
src="In practice.files/img292.png" width=97 border=0> </DIV><BR clear=all>
<P></P>
<P>If we are interested in the behaviour of a random variable <!-- MATH $\widehat{\theta}=\theta({\cal X},F)$ --><IMG height=46 
alt="$\widehat{\theta}=\theta({\cal X},F)$" src="In practice.files/img293.png" 
width=109 align=middle border=0>, then we can consider the sequence of <IMG 
height=17 alt=$B$ src="In practice.files/img199.png" width=20 align=bottom 
border=0> new values obtained through computation of <IMG height=17 alt=$B$ 
src="In practice.files/img199.png" width=20 align=bottom border=0> new bootstrap 
samples. 
<P>Practically speaking this will need generatation of an integer between 1 and 
n, each of these integers having the same probability through a function denoted 
by <I>RANINT(1,n)</I> hereof. 
<P>Here is an example of a line of matlab that does just that: <PRE>T= floor((rand(1)*n)+1);
</PRE>
<P>If we use S we won't need to generate the new observations one by one, the 
following command generates a n-vector with replacement in the vector of indices 
(1...n). 
<P><PRE>sample(n,n,replace=T)
</PRE>
<P>An approximation of the distribution of the estimate <!-- MATH $\widehat{\theta}=\theta({\cal X}_n,F)$ --><IMG height=32 
alt="$\widehat{\theta}=\theta({\cal X}_n,F)$" src="In practice.files/img294.png" 
width=389 align=middle border=0> is provided by the distribution of <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\widehat{\theta}^{*b}=\theta({\cal X}_n^{*b},F_n), \  \ b=1..B\end{displaymath} --><IMG 
height=52 
alt="\begin{displaymath}\widehat{\theta}^{*b}=&#10;\theta({\cal X}_n^{*b},F_n), \ \ b=1..B&#10;\end{displaymath}" 
src="In practice.files/img295.png" width=171 border=0> </DIV><BR clear=all>
<P></P><!-- MATH $G_n^*(t)=P_{F_n}\left(\widehat{\theta}^* \leq t \right)$ --><IMG 
height=23 alt="$G_n^*(t)=P_{F_n}\left(&#10;\widehat{\theta}^* \leq t \right)$" 
src="In practice.files/img296.png" width=22 align=middle border=0> denotes the 
bootstrap distribution of <!-- MATH $\widehat{\theta}^*$ --><IMG height=32 
alt=$\widehat{\theta}^*$ src="In practice.files/img297.png" width=368 
align=bottom border=0>, often approximated by <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\widehat{G}_n^*(t)=\#\{\widehat{\theta}^*\leq t\}/B\end{displaymath} --><IMG 
height=45 
alt="\begin{displaymath}&#10;\widehat{G}_n^*(t)=&#10;\char93 \{\widehat{\theta}^*&#10;\leq t&#10;\}/B\end{displaymath}" 
src="In practice.files/img298.png" width=226 border=0> </DIV><BR clear=all>
<P></P>
<P><I><!-- MATH $\fbox{The Bootstrap Algorithm}$ --><IMG height=34 
alt="\fbox{The Bootstrap Algorithm}" src="In practice.files/img299.png" width=29 
align=middle border=0> </I>
<P>
<OL>
  <LI>For b=1 to B do : /* B is the number of bootstrap samples */ 
  <OL>
    <LI>For i=1 to n do : /* Generation of Sample <IMG height=46 
    alt="${\cal X}_b^*$" src="In practice.files/img300.png" width=95 
    align=middle border=0> */ 
    <OL>
      <LI>OBS=RANINT(1,n) 
      <LI>Add observation number OBS to Sample <IMG height=46 
      alt="${\cal X}_b^*$" src="In practice.files/img300.png" width=95 
      align=middle border=0> </LI></OL>
    <LI>Compute <!-- MATH $\widehat{\theta}^*_b=\theta({\cal X}_b^*)$ --><IMG 
    height=31 alt="$\widehat{\theta}^*_b=\theta({\cal X}_b^*)$" 
    src="In practice.files/img301.png" width=363 align=middle border=0> 
  </LI></OL></LI></OL>
<P>An example data set is LS (or laws) : the law school data consisting in 15 
bivariate data points, each corresponding to an American law school as analysed 
in E<SMALL>FRON 1982</SMALL>: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}x_i=(LSAT_i,GPA_i)\end{displaymath} --><IMG 
height=34 alt="\begin{displaymath}&#10;x_i=(LSAT_i,GPA_i)\end{displaymath}" 
src="In practice.files/img302.png" width=54 border=0> </DIV><BR clear=all>
<P></P><IMG height=34 alt=$LSAT_i$ src="In practice.files/img303.png" width=54 
align=middle border=0> is the class average score on a nationwide exam, <IMG 
height=34 alt=$GPA_i$ src="In practice.files/img304.png" width=26 align=middle 
border=0> is the class average undergraduate grades. 
<P>In order to repeat a large number of times the same instructions needed to 
construct <IMG height=17 alt=$B$ src="In practice.files/img199.png" width=20 
align=bottom border=0> data sets one usually has to use a looping facility, 
actually matlab and S are easier to program in this circumstance because one can 
use ordinary loops and use owner-created functions, as well as trying to benefit 
form the vectorized commands. 
<P>
<HR>
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<ADDRESS>Susan Holmes 2002-01-12 </ADDRESS></BODY></HTML>

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