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<H2><A name=SECTION00314000000000000000>Smoothed Bootstrap</A> </H2>An
intermediate solution between parametric and nonparametric bootsrapping is the
smoothed bootstrap.
<P>Instead of resampling directly from the empirical distribution <IMG height=45
alt=$F_n$ src="Smoothed Bootstrap.files/img309.png" width=26 align=middle
border=0>, one smoothes it out first then the smoothed version is used to
generate the new samples.
<P>If a kernel smoother is used, for instance a Gaussian kernel, then generation
from the smoothed distribution <IMG height=31 alt=$\widehat{F}_n$
src="Smoothed Bootstrap.files/img310.png" width=390 align=middle border=0> is
easy since it is sufficient to resample with replacement and then perturb the
sampled point by adding a small gaussian random variable (S<SMALL>ILVERMAN
</SMALL>B.W. & Y<SMALL>OUNG </SMALL>G.A.)
<P><PRE>#Smoothed Bootstrap
"boots"_function(x,nboot=100,fun,kernel="norm",h=1)
{
#Function doing a smoothed bootstrap with either
# a normal or an epaneichikov kernel
#h is the window width (can be tricky to choose)
n_nrow(as.matrix(x))
theta_rep(0,nboot)
for (k in (1:nboot))
{
if(kernel=="norm") eps_rnorm(n,mean=0,sd=1)
else eps_epan(n)
xb_x[sample(n,n,T)]+eps*h
theta[k]_fun(xb)
}
return(theta)
}
"epan"_function(n)
{
eps_rep(0,n)
v_matrix(runif(3*n,-1,1),nrow=n)
for(i in(1:n))
{
if ((abs(v[i,3])>=abs(v[i,2])) &&
(abs(v[i,3])>=abs(v[i,1])))
eps[i]_v[i,2]
else eps[i]_v[i,3]
}
return(eps)
}
</PRE>
<P><BR>
<HR>
<ADDRESS>Susan Holmes 2002-01-12 </ADDRESS></BODY></HTML>
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