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I am interested in non-parametric methods for building confidence intervals for an estimator (e.g. the mean) using few samples (e.g. 10). I think I have read somewhere that smoothing the bootstrapped estimator values can improve the quality of the derived percentiles interval. However I could not find any online reference that explains how to tune the bandwidth of the smoothing step.

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    $\begingroup$ I am not an expert, but I do not think that smoothed bootstrap always performs better. Anyway, in some cases it might be and a standard smooth estimate of a continuous pdf is the kernel density estimation for which there is a huge amount of literature on how to choose the bandwidth h including rules which make reference to the normal distribution, or cross-validation methods. $\endgroup$
    – ocram
    May 2 '12 at 5:11
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    $\begingroup$ For example, you can find details here: google.be/… $\endgroup$
    – ocram
    May 2 '12 at 5:11
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    $\begingroup$ In line with @ocram's comment, you might be interested on comparing different types of bootstrap intervals. Take a look at this question. $\endgroup$
    – user10525
    May 2 '12 at 7:39
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Sorry to be answering so late. This question came just when I joined CV and i only found it by looking back. For your specific question about using the bootstrap in kernel density estimation I think you will find material in Bernard Silverman's book. I think he covers the use of bootstrap for bandwidth selection.

Efron and Tibshirani discuss the bootstrap for finding modes of a density via kernel methods.

Oddly there is not really much on it in the general text on bootstrap including mine. Maybe in the next edition I will add something.

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    $\begingroup$ I think the OP is asking a slightly different question which is about using kernel density estimation for producing improved confidence intervals when the bootstrap sample is small: "I am interested in non-parametric methods for building confidence intervals for an estimator (e.g. the mean) using few samples (e.g. 10)". $\endgroup$
    – user10525
    Jul 24 '12 at 16:15
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    $\begingroup$ That was my impression as well @Procrastinator (+1). Michael, note that there's no need to apologize for answering late. In fact, I believe that attending to old questions is actually encouraged, as evidenced by the existence of the Necromancer, Revival and Archaeologist badges. $\endgroup$
    – Macro
    Jul 24 '12 at 16:50
  • $\begingroup$ Thanks guys. I know I was not directly answering the question but it was my impression that the OP might have been a little misinformed or misinterpreted what he read. With respect to bootstrap confidence intervals there are many variations that adjust the endpoints. Those I am familiar with but not anything applying a smoothed bootstrap. Of course there is a smoothed bootstrap which involves sampling from a kernel density fit to the original sample. $\endgroup$ Jul 24 '12 at 17:52
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Bootstrap confidence intervals

Thomas J. DiCiccio and Bradley Efron Source: Statist. Sci. Volume 11, Number 3 (1996), 189-228.

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ss/1032280214

Keywords: Bootstrap-t; BCa and ABC methods; calibration; second-order accuracy

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