On all 3 graphs
- Θ is the population parameter, of which is an estimate - calculated from a random sample.
- Areas shaded orange fall outside the intervals I, or I_{min} or ^{I}/_{2}+^{I}/_{2}, all of which enclose (100[1−α] =) 95% of this population of estimates.
- a is the proportion of estimates less than _{(a)}, and b is the proportion of estimates greater than _{(b)}
- _{(a)} and _{(b)} are the lower and upper limits of the (parametric) interval, I.
- Notice that (a) and (b) are the rank of _{(a)} and
_{(b)}
These probability density plots depict a theoretical parametric smooth distribution, rather than the practical situation where you need to obtain confidence limits from the ranks of your bootstrap statistics - in other words, you wish to estimate a 100[1−α]% interval from B bootstrap estimates.
Where B is very large, and ^{*} is smooth:
- To estimate the equal-tailed confidence interval for ,
Î = ^{*}_{(b)} − ^{*}_{(a)}
- Calculate the rank, (a), = Bα/2.
- and the rank, (b), = B[1−α]/2.
Then identify ^{*}_{(a)} and ^{*}_{(b)}
- To estimate the shortest confidence interval for ,
Î_{min} is the smallest value of ^{*}_{(b)} − ^{*}_{(a)}.
For each possible rank, (a),
- calculate the corresponding (b) = (a) + B[1−α]/2.
Then select the pair of bootstrap estimates with the smallest |^{*}_{(b)} − ^{*}_{(a)}|.
- To estimate the symmetrical confidence interval for ,
Î = 2[−^{*}_{(a)}] = 2[^{*}_{(b)}−]
For each possible ^{*}_{(a)},
- calculate ^{*}_{(b)} = 2 − ^{*}_{(a)}
- and calculate = (a)/B + [1 − (b)/B].
Then identify the pair of bootstrap estimates whose = α.
In practice, these confidence limit formulae are only approximate because B is finite and ^{*} is discrete. But, provided ^{*} does not have a strongly stepped distribution, these approximations may be improved using interpolation and mean ranks.