Bootstrap intervals

01Understand the idea#

A percentile bootstrap interval uses the middle part of a collection of bootstrap statistic values. It summarizes uncertainty in the statistic rather than selecting the middle observations from the original dataset. First resample, then compute one statistic per resample, then find the stated percentiles.

A percentile bootstrap interval uses two quantiles of bootstrap statistic outputs to approximate a confidence interval. For a central 90% interval, leave 5% in each tail and read the 5th and 95th percentiles.

For central fraction $c$, tail fractions are $(1-c)/2$ and $(1+c)/2$. Sort the $B$ supplied replicates, retaining repeats. Multiply a tail fraction $p$ by $B$, and round upward to a rank.

If bootstrap means are $2,3,4,5,8$, sort these five values before using a percentile rule. Under a rule that selects the second and fourth ordered values, the interval is $[3,5]$. A different interpolation rule could give different endpoints, so follow the exact quantile convention supplied.

For these eight outputs, central 50% uses $p=0.25,0.75$. Ranks are $2,6$, so the interval is $[2,7]$.

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02The interval need not center on the original estimate#

The interval need not center on the original estimate. In the example, estimate $4$ does not force equal distances to $2$ and $7$. These tiny lists illustrate endpoint mechanics; they do not establish reliable coverage.

03More resamples make the empirical percentiles more stable#

More resamples make the empirical percentiles more stable. They do not add new observations. A small original sample or an extreme statistic, such as a maximum, can give poor coverage even with many resamples.

04A closer look#

A percentile interval is an approximation and can perform poorly for small samples, strong bias, or unsuitable statistics. Many computational replicates reduce simulation noise in the endpoints; they do not repair an unrepresentative original sample.