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 Imin 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
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 ,
Îmin 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.