# How to select the bandwith of smoothed bootstrap for building confidence intervals?

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.

• 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. May 2, 2012 at 5:11
• For example, you can find details here: google.be/… May 2, 2012 at 5:11
• In line with @ocram's comment, you might be interested on comparing different types of bootstrap intervals. Take a look at this question.
– user10525
May 2, 2012 at 7:39

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.

• 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)".
– user10525
Jul 24, 2012 at 16:15
• 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. Jul 24, 2012 at 16:50
• 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. Jul 24, 2012 at 17:52

Bootstrap confidence intervals

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