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"It has long been an axiom of mine that the little things are infinitely the most important" (Sherlock Holmes)

Just a note

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 (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 ,
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.