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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.