Basic bootstrap interval with larger numbers

Question

Your statistic on the original sample is $\hat{\theta} = 50$ (say, a median dollar PnL). Bootstrapping gives a 2.5th percentile of $38$ and a 97.5th percentile of $61$ .

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

Accepted Answer

The basic interval reflects the bootstrap quantiles through twice the estimate: \left[\, 2\hat{	heta} - q{0.975}, \;\; 2\hat{	heta} - q{0.025} \,ight]. Plug in \hat{	heta} = 50, q{0.025} = 38, q{0.975} = 61. 	ext{lower} = 2(50) - 61 = 100 - 61 = 39, 	ext{upper} = 2(50) - 38 = 100 - 38 = 62. So the basic interval is [39, 62]. The percentile interval would have been [38, 61]; the small shift reflects a slight downward lean in the bootstrap distribution (its lower tail reaches 12 below \hat{	heta}, its upper tail only 11 above). Takeaway. The reflection rule is scale-free; only the arith…

Basic bootstrap interval with larger numbers

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