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Chapter 7: Data Partitioning 247

This yields an estimate of the standard error of the median of
(
ˆ
ˆ
SE Boot q 0.5 ) = 7.1 . In the exercises, the reader is asked to see what happens
when the statistic is the mean and should find that the jackknife and boot-
strap estimates of the standard error of the mean are similar.

It can be shown [Efron & Tibshirani, 1993] that the jackknife estimate of the
standard error of the median does not converge to the true standard error as
n → ∞ . For the data set of Example 7.7, we had only two distinct values of
the median in the jackknife replicates. This gives a poor estimate of the stan-
dard error of the median. On the other hand, the bootstrap produces data sets
that are not as similar to the original data, so it yields reasonable results. The
delete-d jackknife [Efron and Tibshirani, 1993; Shao and Tu, 1995] deletes d
observations at a time instead of only one. This method addresses the prob-
lem of inconsistency with non-smooth statistics.

7.4 Better Bootstrap Confidence Intervals

In Chapter 6, we discussed three types of confidence intervals based on the
bootstrap: the bootstrap standard interval, the bootstrap-t interval and the
bootstrap percentile interval. Each of them is applicable under more general
assumptions and is superior in some sense (e.g., coverage performance,
range-preserving, etc.) to the previous one. The bootstrap confidence interval
that we present in this section is an improvement on the bootstrap percentile
interval, which stands for bias-corrected and
interval. This is called the BC a
accelerated.
⋅
Recall that the upper and lower endpoints of the 1 –( α) 100% bootstrap
percentile confidence interval are given by
ˆ * α 2⁄(
⁄
(
ˆ
ˆ
,
,
(
Percentile Interval: θ Lo θ Hi) = ( θ B ) ˆ *1 – α 2) . ) (7.15)
θ B
Say we have B = 100 bootstrap replications of our statistic, which we denote
ˆ *b
,
as θ , b = 1 … 100 . To find the percentile interval, we sort the bootstrap
,
replicates in ascending order. If we want a 90% confidence interval, then one
ˆ
way to obtain θ Lo is to use the bootstrap replicate in the 5th position of the
ˆ
ordered list. Similarly, θ Hi is the bootstrap replicate in the 95th position. As
discussed in Chapter 6, the endpoints could also be obtained using other
quantile estimates.
interval adjusts the endpoints of the interval based on two param-
The BC a
ˆ ˆ . The 1 –( ⋅
a
eters, and z 0 α) 100% confidence interval using the BC a
method is
© 2002 by Chapman & Hall/CRC