Page 235 - Computational Statistics Handbook with MATLAB

222 Computational Statistics Handbook with MATLAB

ˆ
ˆ
ˆ
⁄
( θ – ˆ 1 –( α 2) ⋅ SE θ θ –, ˆ ˆ α 2⁄( ) ⋅ SE θ ˆ , ) (6.24)
t
t
ˆ
ˆ ˆ
θ
where SE is an estimate of the standard error of . The bootstrap-t interval
is suitable for location statistics such as the mean or quantiles. However, its
accuracy for more general situations is questionable [Efron and Tibshirani,
1993]. The next method based on the bootstrap percentiles is more reliable.
PROCEDURE - BOOTSTRAP-T CONFIDENCE INTERVAL
,
,
1. Given a random sample, x = ( x 1 … x n ) , calculate . θ ˆ
2. Sample with replacement from the original sample to get
,
b b b
,
*
x * = ( x 1 … x n * . )
3. Calculate the same statistic using the sample in step 2 to get θ ˆ *b .
*b
4. Use the bootstrap sample x to get the standard error of θ ˆ *b . This
can be calculated using a formula or estimated by the bootstrap.
5. Calculate z *b using the information found in steps 3 and 4.
6. Repeat steps 2 through 5, B times, where B ≥ 1000 .
⁄
7. Order the z *b from smallest to largest. Find the quantiles t ˆ 1 –( α 2)
and t ˆ α 2⁄( ) .
ˆ ˆ
θ
8. Estimate the standard error SE θ ˆ of using the B bootstrap repli-
cates of θ ˆ *b (from step 3).
9. Use Equation 6.24 to get the confidence interval.
The number of bootstrap replicates that are needed is quite large for confi-
dence intervals. It is recommended that B should be 1000 or more. If no for-
mula exists for calculating the standard error of θ ˆ *b , then the bootstrap
method can be used. This means that there are two levels of bootstrapping:
ˆ *b
one for finding the SE and one for finding the z *b , which can greatly
increase the computational burden. For example, say that B = 1000 and we
ˆ *b
use 50 bootstrap replicates to find SE , then this results in a total of 50,000
resamples.

Example 6.11
Say we are interested in estimating the variance of the forearm data, and we
decide to use the following statistic,

n
ˆ 2
---
σ = 1 ∑ ( X i – X) 2 ,
n
i = 1

© 2002 by Chapman & Hall/CRC