📄 bias.htm
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<H4><A name=SECTION00315110000000000000>Bias </A></H4>: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}BIAS=E_F(\theta(F_n))-\theta(F)\end{displaymath} --><IMG
height=37
alt=\begin{displaymath}BIAS=E_F(\theta(F_n))-\theta(F)\end{displaymath}
src="Bias.files/img311.png" width=199 border=0> </DIV><BR clear=all>
<P></P>If we take the statistic <!-- MATH $R({\cal X},F)= \theta(F_n)-\theta(F)$ --><IMG height=64
alt="$R({\cal X},F)= \theta(F_n)-\theta(F)$" src="Bias.files/img312.png"
width=455 align=middle border=0>, (where <IMG height=45 alt=$F_n$
src="Bias.files/img309.png" width=26 align=middle border=0> denotes the
empirical distribution function) and apply the bootstrap algorithm to its
expectation then we obtain the estimate : <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\widehat{BIAS}=E_*R^*=\frac{1}{B}\sum_{b=1}^B(\widehat{\theta}^{*b}-\widehat{\theta})=\widehat{\theta}^{*\bullet}-\widehat{\theta}\end{displaymath} --><IMG
height=46
alt="\begin{displaymath} \widehat{BIAS}=E_*R^*= \frac{1}{B}\sum_{b=1}^B(\widehat{\the... ...widehat{\theta}) =\widehat{\theta}^{*\bullet}-\widehat{\theta} \end{displaymath}"
src="Bias.files/img313.png" width=141 border=0> </DIV><BR clear=all>
<P></P>where the dot indicates averaging :<!-- MATH $\widehat{\theta}^{*\bullet}=\sum_{b=1}^B\widehat{\theta}^{*b}/B$ -->
<IMG height=70
alt="$\widehat{\theta}^{*\bullet}= \sum_{b=1}^B\widehat{\theta}^{*b}/B$"
src="Bias.files/img314.png" width=410 align=middle border=0> <BR><BR>
<HR>
<ADDRESS>Susan Holmes 2002-01-12 </ADDRESS></BODY></HTML>
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