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I have seen that a number of Statistics texts state that in order to get the $1-a$ confidence interval, we have to arrange our $B$ bootstap sample estimators in order statistics:

$$ \theta_{(1)} \leq \theta_{(2)} \leq \ldots \leq \theta_{(B)} $$

Then choose $m= \left[ \left( a/2 \right) B \right]$ where $[.]$ denotes the greatest integer value and $a$ is the desired significance level. Then the interval:

$$ \left( \theta_{(m)}, \theta_{(B+1-m)} \right) $$

gives the desired CI.

I understand the lower value, what I do not seem to understand is the upper limit of the interval, especially why we need to add 1. I would think it was like $B(1-a/2)=B-m$ but that 1 there confuses me.

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  • $\begingroup$ This is due to the right-continuity of (empirical) distribution functions and the fact that statisticians tend to choose the conservative option if in doubt. Please note that percentile bootstrap CIs are often not very appropriate (e.g. if the bootstrap distribution looks rather asymmetric). $\endgroup$ – Michael M Nov 28 '13 at 18:27
  • $\begingroup$ @MichaelMayer Yes indeed, the distribution is often skewed. But why is the right continuity important here? Thanks. $\endgroup$ – JohnK Nov 28 '13 at 18:31
  • $\begingroup$ The bootstrap percentile CI is defined by appropriate low and high quantiles of the bootstrap distribution. Inversion of such jump function is not very clearly defined, that is why there are so many different ways to compute sample quantiles. In principle you can choose between all possibilities. $\endgroup$ – Michael M Nov 28 '13 at 18:37
  • $\begingroup$ @MichaelMayer It is not wrong then to drop the 1 that confuses me for the sake of a simpler alternative? $\endgroup$ – JohnK Nov 28 '13 at 18:41
  • $\begingroup$ No, I wouldn't invent a new method to compute sample quantiles. $\endgroup$ – Michael M Nov 28 '13 at 18:46

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