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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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    $\begingroup$ in my opinion, mean of bootstraps $\endgroup$ – FMZ Mar 16 '12 at 1:40
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    $\begingroup$ 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). $\endgroup$ – whuber Apr 3 '12 at 5:59
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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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The observed value has the highest likelihood and is usually considered to be the best estimate. Bootstrapping is a method for estimating variability, but it doesn't improve on the observed estimate of the parameter. The bootstrap estimate of the parameter should asymptote to the observed value with enough resamples...

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    $\begingroup$ Peter Hall disagrees with every assertion in this reply. One of the emphases in his The Bootstrap and Edgeworth Expansion (Springer-Verlag 1992) is on bias reduction in parameter estimates using the bootstrap. This shows that (a) it is for more than estimating variability and (b) definitely has a use for improving parameter estimates. Finally, most estimates (including the bootstrap) will not reach some particular "observed value" asymptotically (this, after all, is just one random draw from a distribution): instead, they will approach the true value asymptotically. $\endgroup$ – whuber Apr 3 '12 at 5:56
  • $\begingroup$ @whuber I have not read that book, but a quick simulation in Excel suggests that my assertion is correct. A sample of ten values from N(0,1) gave mean=-0.435 and stdev = 0.73. 10240 bootstrap resamples of the 10 observations gave an average mean of -0.435 and an average stdev of 0.684. How could resampling yield a long-run mean different to the original sample mean? $\endgroup$ – Michael Lew Apr 3 '12 at 23:54
  • $\begingroup$ I'm sorry Michael, I don't see any kind of calculations in this reply, so I don't understand how an Excel simulation is relevant here. Your reply makes three assertions about bootstrapping: (1) it is for estimating variability--no, that's too limiting; (2) doesn't improve on the estimate--on the contrary, one of its best uses is in bias reduction; and (3) something about its asymptotic behavior that simply makes no sense. In addition, you make a claim about an observed value having "the highest likelihood"; this, too, makes no sense, because likelihood is a function of the parameters. $\endgroup$ – whuber Apr 4 '12 at 12:42
  • $\begingroup$ @whuber I don't understand. How can it be that resampling a sample will on average give a different mean to the mean of the original sample? In the bootstraps that I ran with Excel the mean of the bootstrap means was exactly the same as the mean of the original sample, as I expected. Is that not relevant? With repect to the likelihood I was a bit loose in language, but the parameter value with the highest likelihood is the value of the parameter equal to the observed estimate. Surely that is generally correct... $\endgroup$ – Michael Lew Apr 5 '12 at 6:34
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    $\begingroup$ I believe I outlined an example of bootstrap bias reduction in my preceding comment, Michael, which is a form of improvement in an estimate. If you would like to pose this as a question, I will endeavor to provide a more detailed reply. In the meantime, you may consult the first chapter of Hall's book or Chapter 10 of the Efron & Tibshirani book (An Introduction to the Bootstrap, Chapman & Hall, 1993). $\endgroup$ – whuber Apr 6 '12 at 15:57

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