The basic (reverse-percentile) bootstrap interval

Question

Your statistic on the original sample is $\hat{\theta} = 5.0$ . You bootstrap it and find that the resampled values have a 2.5th percentile of $4.2$ and a 97.5th percentile of $6.4$ .

Construct the 95% basic (reverse-percentile) bootstrap confidence interval.

Accepted Answer

The percentile interval would just read the endpoints straight off the bootstrap quantiles: [4.2, 6.4]. The basic interval starts from a different (and often better-justified) idea: it looks at how the bootstrap statistic deviates from \hat{	heta}, and reflects those deviations back through \hat{	heta}. If q{0.025} and q{0.975} are the bootstrap quantiles, the basic interval is \left[\, 2\hat{	heta} - q{0.975}, \;\; 2\hat{	heta} - q{0.025} \,ight]. Notice the quantiles swap ends: the upper bootstrap quantile sets the lower limit. Plug in \hat{	heta} = 5.0, q{0.025} = 4.2, q{0.975} = 6.…

The basic (reverse-percentile) bootstrap interval

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