# Nonparametric bootstrap confidence interval

I have generated a 95% confidence interval $(22,25)$ for a parameter using a nonparametric bootstrapping method. What I want to know is what value we use for the estimate of the parameter.

Is it the mean of all the bootstraps, the median of all the bootstraps, the original estimate before we bootstrapped from the original dataset, or something else?

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in my opinion, mean of bootstraps –  FMZ Mar 16 '12 at 1:40
One correct answer that seems to have gone unmentioned in the replies so far is indeed "something else." See, for instance, chapter 2 of The Bootstrap Small Sample Properties (F. W. Scholz 2007). –  whuber Apr 3 '12 at 5:59

Taking the mean of your bootstrap distribution is called bagging (from bootstrap aggregating; link). I've never seen it used on parameters, just on predictions, but it has a lot in common with Bayesian model averaging, which can work well on parameters. in this framework, your parameter estimated from the original data is like your posterior mode and the bagged estimate is like your posterior mean. The posterior mean often has better accuracy out of sample, but I'm not sure that applies to your case.

A few things to consider:

• Does the mean of your bootstrap distribution look like your maximum likelihood estimate? If so, it might not matter which you choose.

• Can you try it both ways on a subset of your data and see which works better on a validation set?

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Bootstrapping is useful for measuring the variability of sample estimates. I report my sample estimate as well as the bootstrapped confidence interval. Wikipedia appears to agree: "bootstrapping is a method for assigning measures of accuracy to sample estimates."

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