Page 233 - Computational Statistics Handbook with MATLAB

220 Computational Statistics Handbook with MATLAB

2. Sample with replacement from the original sample to get
* b
,
,
x * b = ( x 1 … x n * b . )
3. Calculate the same statistic using the sample in step 2 to get the
bootstrap replicates, θ ˆ *b .
4. Repeat steps 2 through 3, B times.
5. Using the bootstrap replicates, calculate θ ˆ * .
θ
ˆ
6. Estimate the bias of using Equation 6.17.
Example 6.10
We return to the forearm data of Example 6.9, where now we want to esti-
mate the bias in the sample skewness. We use the same bootstrap replicates
as before, so all we have to do is to calculate the bias using Equation 6.17.
% Use the same replicates from before.
% Evaluate the mean using Equation 6.15.
meanb = mean(thetab);
% Now estimate the bias using Equation 6.17.
biasb = meanb - theta;
We have an estimated bias of -0.011. Note that this is small relative to the stan-
dard error.

In the next chapter, we discuss another method for estimating the bias and
the standard error of a statistic called the jackknife. The jackknife method is
related to the bootstrap. However, since it is based on the reuse or partition-
ing of the original sample rather than resampling, we do not include it here.

onfide
onfiden
Bootstrap CC onfideonfide nnceInteInte rvval als ss s
C
BootstrapC
ce
nceInteceInte
BootstrapBootstrap
rr vvalal
r
There are several ways of constructing confidence intervals using the boot-
strap. We discuss three of them here: the standard interval, the bootstrap-t
interval and the percentile method. Because it uses the jackknife procedure,
will be presented
an improved bootstrap confidence interval called the BC a
in the next chapter.
ddeenceIntervaInterva
ii
a
nce
BootstrapStand
BootstrapStandStand
d
rdConfdConfi
Bootstrap
Bootstrap Stand ar aarr dConfConf idde enceIntervanceInterva l ll l
The bootstrap standard confidence interval is based on the parametric form
of the confidence interval that was discussed in Section 6.2. We showed that
⋅
the 1 –( α) 100% confidence interval for the mean can be found using
 ( 1 – α 2) σ ( α 2) σ 
⁄
⁄
P X – z ------- < µ < X – z -------  = 1 – α . (6.20)
 n n 
© 2002 by Chapman & Hall/CRC