Basic versus percentile bootstrap interval

Question

Your statistic on the original sample is $\hat{\theta} = 2.0$ . The bootstrap distribution is strongly right-skewed: its 2.5th percentile is $1.9$ (just below the estimate) and its 97.5th percentile is $3.2$ (far above it).

Construct the 95% basic (reverse-percentile) bootstrap confidence interval, and note how it compares with the percentile interval.

Accepted Answer

The percentile interval reads straight off the quantiles: [1.9, 3.2]. But look where \hat{	heta} = 2.0 sits: almost at the bottom of that range, because the bootstrap distribution has a long upper tail. The percentile interval barely extends below the estimate, which is suspicious. The basic interval reflects the quantiles through 2\hat{	heta}: \left[\, 2\hat{	heta} - q{0.975}, \;\; 2\hat{	heta} - q{0.025} \,ight]. Plug in \hat{	heta} = 2.0, q{0.025} = 1.9, q{0.975} = 3.2. 	ext{lower} = 2(2.0) - 3.2 = 4.0 - 3.2 = 0.8, 	ext{upper} = 2(2.0) - 1.9 = 4.0 - 1.9 = 2.1. So the basic interva…

Basic versus percentile bootstrap interval

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