Page 237 - Computational Statistics Handbook with MATLAB

224 Computational Statistics Handbook with MATLAB

Now we get the quantiles that we need for the interval given in Equation 6.24
and calculate the interval.

% Get the quantiles.
k = B*alpha/2;
szval = sort(zvals);
tlo = szval(k);
thi = szval(B-k);
% Get the endpoints of the interval.
blo = thetahat - thi*SE;
bhi = thetahat - tlo*SE;
The bootstrap-t interval for the variance of the forearm data is 1.00 1.57,( . )

Bootstrap
BootstrapPercen nt nntt tileIntervaInterva l ll
BootstrapPercePerce
Perce
Bootstrap
ile
ile
ileIntervaInterva
l
An improved bootstrap confidence interval is based on the quantiles of the
distribution of the bootstrap replicates. This technique has the benefit of
being more stable than the bootstrap-t, and it also enjoys better theoretical
coverage properties [Efron and Tibshirani, 1993]. The bootstrap percentile
confidence interval is
⁄
ˆ * α 2⁄(
(
,
( θ B ) ˆ *1 – α 2) , ) (6.25)
θ B
ˆ * α 2⁄( ) is the α 2⁄ quantile in the bootstrap distribution of θ ˆ *
where θ B . For
(
example, if α 2⁄ = 0.025 and B = 1000 , then θ B * 0.025) is the θ ˆ *b in the 25th
ˆ
ˆ * 0.975( )
position of the ordered bootstrap replicates. Similarly, θ B is the replicate
in position 975. As discussed previously, some other suitable estimate for the
quantile can be used.
The procedure is the same as the general bootstrap method, making it easy
to understand and to implement. We outline the steps below.
PROCEDURE - BOOTSTRAP PERCENTILE 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
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, where B ≥ 1000 .
5. Order the θ ˆ *b from smallest to largest.
⁄
6. Calculate B α 2⁄⋅ and B (⋅ 1 – α 2) .

© 2002 by Chapman & Hall/CRC