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border=0></A> <BR><B>Next:</B> <A
href="http://www-stat.stanford.edu/~susan/courses/s208/node16.html"
name=tex2html356>The jackknife</A> <B>Up:</B> <A
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name=tex2html354>Lectures</A> <B>Previous:</B> <A
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name=tex2html348>Monte Carlo</A> <BR><BR><!--End of Navigation Panel--><!--Table of Child-Links--><A
name=CHILD_LINKS><STRONG>Subsections</STRONG></A>
<UL>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00291000000000000000"
name=tex2html357>Statistical Functionals</A>
<UL>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00291100000000000000"
name=tex2html358>Notions of Convergence</A>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00291200000000000000"
name=tex2html359>Why is the empirical cdf <IMG height=45 alt=$\hat{F}_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img16.png"
width=26 align=middle border=0> a good estimator of F?</A>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00291300000000000000"
name=tex2html360>Generalized Statistical Functionals</A> </LI></UL><BR>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00292000000000000000"
name=tex2html361>Example and Counterexample</A>
<UL>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00292100000000000000"
name=tex2html362>Bootstrap of the maximum</A>
<UL>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00292110000000000000"
name=tex2html363>Theoretical Analysis</A> </LI></UL></LI></UL><BR>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00293000000000000000"
name=tex2html364>Parametric Bootstrap</A>
<UL>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00293100000000000000"
name=tex2html365>Maximum</A>
<LI><A
href="http://www-stat.stanford.edu/~susan/courses/s208/node15.html#SECTION00293200000000000000"
name=tex2html366>Correlation Coefficient</A> </LI></UL></LI></UL><!--End of Table of Child-Links-->
<HR>
<H1><A name=SECTION00290000000000000000>More about the theoretical underpinnings
of the Bootstrap</A> </H1>
<P>
<H2><A name=SECTION00291000000000000000>Statistical Functionals</A> </H2>
<P>(Reference : Eric Lehmann, 1998,pp.381-438.) <BR>We often speak of the
asymptotic properties of the sample mean <IMG height=19 alt=$\bar{X}$
src="More about the theoretical underpinnings of the Bootstrap.files/img12.png"
width=22 align=bottom border=0>.These refer to the sequence <IMG height=40
alt=$\bar{X}_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img173.png"
width=29 align=middle border=0>. These functions are the <I>same</I> in some
sense, for all sample size. The notion of statistical functional makes this
clearer.
<P>Suppose we are interested in real-valued parameters. We often have a
situation where the parameter of interest is a function of the distribution
function <IMG height=16 alt=$F$
src="More about the theoretical underpinnings of the Bootstrap.files/img1.png"
width=19 align=bottom border=0>, these are called statistical functionals. <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\theta=s(F)\end{displaymath} --><IMG
height=33 alt=\begin{displaymath}\theta=s(F)\end{displaymath}
src="More about the theoretical underpinnings of the Bootstrap.files/img174.png"
width=70 border=0> </DIV><BR clear=all>
<P></P>Examples: <BR><BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\mu=E_F(X),\qquad \mu^{(k)}=E_F(X-E(X))^k , \qquad F^{-1}(p)\end{displaymath} --><IMG
height=33
alt="\begin{displaymath}\mu=E_F(X),\qquad \mu^{(k)}=E_F(X-E(X))^k , \qquad F^{-1}(p) \end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img175.png"
width=428 border=0> </DIV><BR clear=all>
<P></P>Goodness of fit statistics: <BR><BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\mbox{Kolmogorov-Smirnov 's } h(F)=sup_x |F(x)-F_0(x)|\end{displaymath} --><IMG
height=33
alt="\begin{displaymath}\mbox{Kolmogorov-Smirnov 's } h(F)=sup_x \vert F(x)-F_0(x)\vert \end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img176.png"
width=416 border=0> </DIV><BR clear=all>
<P></P>is estimated by: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}h(\mbox{$\hat{F}_n$})=sup_x |\mbox{$\hat{F}_n$}(x)-F_0(x)|\end{displaymath} --><IMG
height=33
alt="\begin{displaymath}h(\mbox{$\hat{F}_n$})=sup_x \vert\mbox{$\hat{F}_n$}(x)-F_0(x)\vert \end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img177.png"
width=231 border=0> </DIV><BR clear=all>
<P></P>Ratio of two means. <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\theta=\frac{\mu_1}{\mu_2}=\frac{E_{F_1}(X)}{E_{F_2}(X)}\end{displaymath} --><IMG
height=52
alt=\begin{displaymath}\theta=\frac{\mu_1}{\mu_2}=\frac{E_{F_1}(X)}{E_{F_2}(X)}\end{displaymath}
src="More about the theoretical underpinnings of the Bootstrap.files/img178.png"
width=146 border=0> </DIV><BR clear=all>
<P></P>We use the sample cdf <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\hat{F}_n=\frac{\#X_i \leq x}{n}=\frac{1}{n}\sum_{i=1}^n\delta_{\{X_i\leq x\}}\end{displaymath} --><IMG
height=55
alt="\begin{displaymath}\hat{F}_n=\frac{\char93 X_i \leq x}{n}=\frac{1}{n}\sum_{i=1}^n \delta_{\{X_i\leq x\}}\end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img179.png"
width=246 border=0> </DIV><BR clear=all>
<P></P>as the nonparametric estimate of the unknown distribution <IMG height=16
alt=$F$
src="More about the theoretical underpinnings of the Bootstrap.files/img1.png"
width=19 align=bottom border=0>.
<P>The usual estimates for these functionals are obtained by simply plugging in
the empirical distribution function for the unknown theoretical one.
<P>Thus taking into account that for any function <IMG height=33 alt=$g$
src="More about the theoretical underpinnings of the Bootstrap.files/img180.png"
width=14 align=middle border=0> we have: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\int g(x)d\mbox{$\hat{F}_n$}(x)=\frac{1}{n}\sum_{i=1}^n g(x_i)\end{displaymath} --><IMG
height=55
alt="\begin{displaymath}\int g(x)d\mbox{$\hat{F}_n$}(x)=\frac{1}{n}\sum_{i=1}^n g(x_i)\end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img181.png"
width=216 border=0> </DIV><BR clear=all>
<P></P>the plug-in estiamte for the mean is <!-- MATH $\int x dF_n(x) =\frac{1}{n}\sum_{i=1}^n x_i$ --><IMG height=40
alt="$\int x dF_n(x) =\frac{1}{n}\sum_{i=1}^n x_i$"
src="More about the theoretical underpinnings of the Bootstrap.files/img182.png"
width=184 align=middle border=0> the usual sample estimate, for the variance, it
is the biased estimate: <BR>
<P></P>
<DIV align=center><!-- MATH \begin{displaymath}\hat{var_F(X)}=\int (x-E_{\mbox{$\hat{F}_n$}}(x))^2=\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2\end{displaymath} --><IMG
height=55
alt="\begin{displaymath}\hat{var_F(X)}=\int (x-E_{\mbox{$\hat{F}_n$}}(x))^2=\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2 \end{displaymath}"
src="More about the theoretical underpinnings of the Bootstrap.files/img183.png"
width=366 border=0> </DIV><BR clear=all>
<P></P>
<H3><A name=SECTION00291100000000000000>Notions of Convergence</A> </H3><FONT
color=#ff0000>Convergence in Law</FONT> <BR>A sequence of cumulative
distribution functions <IMG height=35 alt=$H_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img184.png"
width=29 align=middle border=0> is said to converge in distribution to <IMG
height=16 alt=$H$
src="More about the theoretical underpinnings of the Bootstrap.files/img185.png"
width=22 align=bottom border=0> iff
<!-- MATH $H_n(x)\longrightarrow H(x)$ --><IMG height=37
alt="$H_n(x)\longrightarrow H(x)$"
src="More about the theoretical underpinnings of the Bootstrap.files/img186.png"
width=138 align=middle border=0> on all continuity points of <IMG height=16
alt=$H$
src="More about the theoretical underpinnings of the Bootstrap.files/img185.png"
width=22 align=bottom border=0>.
<P>We say that if the random variable <IMG height=35 alt=$Y_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img187.png"
width=24 align=middle border=0> has cdf <IMG height=35 alt=$H_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img184.png"
width=29 align=middle border=0> and the rv <IMG height=16 alt=$Y$
src="More about the theoretical underpinnings of the Bootstrap.files/img163.png"
width=20 align=bottom border=0> has cdf <IMG height=16 alt=$H$
src="More about the theoretical underpinnings of the Bootstrap.files/img185.png"
width=22 align=bottom border=0>, <IMG height=35 alt=$Y_n$
src="More about the theoretical underpinnings of the Bootstrap.files/img187.png"
width=24 align=middle border=0> <I>converges in law</I> to <IMG height=16
alt=$Y$
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