# Shifting bootstrap confidence interval to be centered around original parameter

I've been doing a bit of research into bootstrapping as I've been told one method of performing it, and this seems to differ from what I can find in other sources.

I have a sample, and want to estimate the mean or median. I generate 1000 resamples, without replacement, calculating the mean/median for each. By taking the 26th largest and 26th smallest mean/median I calculate, I get a confidence interval.

The sources I've read seem to leave it at that. However, the method I was told went one step further - it calculates the width of this interval, then generates a new interval that is centered around the mean/median of my original sample.

So, if my original mean was 20, and bootstrapping gives me an interval of [17,27], it shifts this to give a final confidence interval of [15,25].

Does this make sense / have any statistical backing? Or is this a mistake?

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You should not be too worried about the symmetry of the interval but about its coverage properties. Asymmetry of the Bootstrap distribution is common when the sample size (of the original distribution) is small/medium. Please, have a look at this paper about the construction of several types of Bootstrap confidence intervals. The ones you are constructing are sort of "quantile-type" CI, but it is better to calculate, say, the $0.025$ and $0.975$ empirical quantiles and construct the corresponding 95% interval. –  user10525 Jun 14 '12 at 7:32

As Procrastinator points out through the reference there are very many ways to generate bootstrap confidence intervals. The method you described is called Efron's percentile method. All bootstrap confidence intervals are approximate in term of the actual coverage. The referenced paper by Efron and DiCiccio describes several of them. Another paper that I recommend that explains the differences between the various bootstrap method is another article in Statistical Science by Efron and Tibshirani that appeared in 1986. They have a table that is particularly good at showing when the various methods work best. I go over this in some detail in my book "Bootstrap Methods: A Guide for Practitioners and Researchers, 2nd Edition" (2007).

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+1 In your experience, what kind of Bootstrap interval has given you better results? –  user10525 Jun 14 '12 at 10:57
There are proofs that show that the higher order bootstraps BCa, bootstrap t and double bootstrap are more accurate in small samples than say the percentile method. However this is not always the case. There are parameters and population distributions where the percentile method works better. LaBudde and I have shown this through simulations when the parameter is the variance and the population distribution is lognormal. –  Michael Chernick Jun 14 '12 at 11:15
Our paper appeared in the American Journal of Mathematical and Management Science (2010). Here is a link: discover-decouvrir.cisti-icist.nrc-cnrc.gc.ca/eng/article/…. –  Michael Chernick Jun 14 '12 at 11:15
Thank you Procrastinator for adding those links to my answer. –  Michael Chernick Jun 14 '12 at 11:17