confidence intervals the percentile method.htm

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<H4><A name=SECTION00315130000000000000>Confidence Intervals : The percentile 
method</A> </H4>As before we denote by <!-- 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="Confidence Intervals  The percentile method.files/img296.png" width=22 
align=middle border=0> the bootstrap distribution of <!-- MATH $\widehat{\theta}^*$ --><IMG height=32 alt=$\widehat{\theta}^*$ 
src="Confidence Intervals  The percentile method.files/img297.png" width=368 
align=bottom border=0>, 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="Confidence Intervals  The percentile method.files/img298.png" width=226 
border=0> </DIV><BR clear=all>
<P></P>
<P>The percentile method consists in taking the <IMG height=40 alt=$1-2\alpha$ 
src="Confidence Intervals  The percentile method.files/img316.png" width=372 
align=middle border=0> confidence interval for <IMG height=17 alt=$\theta$ 
src="Confidence Intervals  The percentile method.files/img19.png" width=14 
align=bottom border=0> as being <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\left[\widehat{G}_n^{*-1}(\alpha),\widehat{G}_n^{*-1}(1-\alpha)\right]\end{displaymath} --><IMG 
height=46 
alt="\begin{displaymath}\left[&#10;\widehat{G}_n^{*-1}(\alpha),&#10;\widehat{G}_n^{*-1}(1-\alpha)&#10;\right]\end{displaymath}" 
src="Confidence Intervals  The percentile method.files/img317.png" width=169 
border=0> </DIV><BR clear=all>
<P></P>Theoretically speaking this is equivalent to replacement of the unknown 
distribution <!-- MATH $G(x,F)=P_F(\widehat{\theta}<x)$ --><IMG height=37 
alt="$G(x,F)=P_F(\widehat{\theta}<x)$" 
src="Confidence Intervals  The percentile method.files/img318.png" width=74 
align=middle border=0> by the estimate <IMG height=45 alt=$G(x,F_n)$ 
src="Confidence Intervals  The percentile method.files/img319.png" width=236 
align=middle border=0>. <BR><I><!-- MATH $\fbox{The Percentile Algorithm }$ --><IMG height=18 
alt="\fbox{The Percentile Algorithm }" 
src="Confidence Intervals  The percentile method.files/img320.png" width=46 
align=middle border=0> : <BR></I>
<P>
<OL>
  <LI>Input the level <IMG height=34 alt=$=2\alpha$ 
  src="Confidence Intervals  The percentile method.files/img321.png" width=25 
  align=bottom border=0> of the confidence interval 
  <LI>For b=1 to B do /* B is the number of bootstrap samples*/ 
  <LI>For i=1 to n do /* Generation of Sample <IMG height=46 alt="${\cal X}_b$" 
  src="Confidence Intervals  The percentile method.files/img322.png" width=91 
  align=middle border=0> */ 
  <OL>
    <LI>OBS=RANINT(1,n) 
    <LI>Add observation number OBS to the Sample <IMG height=46 
    alt="${\cal X}_b$" 
    src="Confidence Intervals  The percentile method.files/img322.png" width=91 
    align=middle border=0> </LI></OL>
  <LI>Compute <!-- MATH $\widehat{\theta}^*_b=\theta({\cal X}_b)$ --><IMG 
  height=37 alt="$\widehat{\theta}^*_b=\theta({\cal X}_b)$" 
  src="Confidence Intervals  The percentile method.files/img323.png" width=64 
  align=middle border=0> 
  <LI>Combine the <!-- MATH $\widehat{\theta}^*_b=\theta({\cal X}_b)$ --><IMG 
  height=37 alt="$\widehat{\theta}^*_b=\theta({\cal X}_b)$" 
  src="Confidence Intervals  The percentile method.files/img323.png" width=64 
  align=middle border=0> into a new data set THETA 
  <LI>Compute the <IMG height=18 alt=$\alpha$ 
  src="Confidence Intervals  The percentile method.files/img33.png" width=17 
  align=bottom border=0> and <IMG height=34 alt=$(1-\alpha)$ 
  src="Confidence Intervals  The percentile method.files/img324.png" width=28 
  align=middle border=0>'th quantiles for THETA. </LI></OL><BR>
<HR>

<ADDRESS>Susan Holmes 2002-01-12 </ADDRESS></BODY></HTML>

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