Page 228 - Computational Statistics Handbook with MATLAB

Chapter 6: Monte Carlo Methods for Inferential Statistics 215

,
,
,,
,
,
x *1 = ( x x x x ) = ( 22 85)
4
4
1
2
,
,
,,,
,
x *2 = ( x 4 x 2 x 3 x 4 ) = ( 2 832).
,
,
We use the notation x *b , b = 1 … B for the b-th bootstrap data set.
In many situations, the analyst is interested in estimating some parameter
θ by calculating a statistic from the random sample. We denote this estimate
by
(
,
,
ˆ
θ = T = t x 1 … x n . ) (6.11)
ˆ
θ
We might also like to determine the standard error in the estimate and the
bias. The bootstrap method can provide an estimate of this when analytical
methods fail. The method is also suitable for situations when the estimator
ˆ t x()
θ = is complicated.
To get estimates of bias or standard error of a statistic, we obtain B boot-
strap samples by sampling with replacement from the original sample. For
every bootstrap sample, we calculate the same statistic to obtain the boot-
ˆ
θ
strap replications of , as follows
,
θ ˆ *b = t x ( *b ); b = 1 … B . (6.12)
,
These B bootstrap replicates provide us with an estimate of the distribution
θ
ˆ
of . This is similar to what we did in the previous section, except that we are
not making any assumptions about the distribution for the original sample.
Once we have the bootstrap replicates in Equation 6.12, we can use them to
understand the distribution of the estimate.
The steps for the basic bootstrap methodology are given here, with detailed
θ
ˆ
procedures for finding specific characteristics of provided later. The issue
of how large to make B is addressed with each application of the bootstrap.
PROCEDURE - BASIC BOOTSTRAP
,
,
1. Given a random sample, x = ( x 1 … x n ) , calculate . θ ˆ
2. Sample with replacement from the original sample to get
,
* b
x * b = ( x , … x * b . )
1
n
3. Calculate the same statistic using the bootstrap sample in step 2 to
get, θ ˆ *b .
4. Repeat steps 2 through 3, B times.
θ
ˆ
5. Use this estimate of the distribution of (i.e., the bootstrap repli-
cates) to obtain the desired characteristic (e.g., standard error, bias
or confidence interval).
© 2002 by Chapman & Hall/CRC