Bootstrap confidence intervals - how many replications to choose? I applied a bootstrap-process to calculate confidence intervalls for the paramters of a multiple lineare regression.
In R it's pretty simple to implement (functions: 'boot' and 'boot.ci') but I still have two comprehension problems:


*

*Why does it make sense to perform a bootstrap procedure before calculating the confidence intervals? Will they be more precise? And if so, can anyone explain why? 

*How can I decide which number of replications is a good number for calculating confidence intervalls? 100? 1000? 10000? 


I would really appreciate any help! 
 A: 
Why does it make sense to perform a bootstrap procedure before calculating the confidence intervals? Will they be more precise? And if so, can anyone explain why? 

You can calculate bootstrap confidence intervals for complex situations, i.e. properties ("statistics") that are not easily accessible analytically. I'm thinking of things like bootstrapping generalization error of a predictive model*.
In other words, bootstrapping may still be possible in situations where you have no good assumption which distribution to base your confidence intervals on. 
The choice parametric (analytical confidence interval based on known distribution) vs. non-parametric bootstrap is a trade-off:


*

*good parametric statistics will be more precise. But they may be totally off if the assumptions are violated (i.e. the distribution you chose was not appropriate).

*bootstrap is less precise (for a given number of original cases) but does not rely on particular distribution assumptions, so there's less danger of getting that part wrong*. 



How can I decide which number of replications is a good number for calculating confidence intervalls? 100? 1000? 10000? 

@MartenBuuis already gave you some idea how to approach this question. Here's another, very pragmatic one:


*

*Bootstrap, say, with nboot = 100 replications. 

*repeat this 10 times

*check variability of the bootstrap results.

*if the variation you observe over the repetitions of the bootstrapping calculation is acceptable for your application, fuse the 10x100 calculations and use the result of that nboot = 10x100 = 1000 replications.
If they are not sufficiently precise, fuse the 10x100 calculations, go back to step 1 and 2 with nboot = 1000 replications.


You get the idea.
A: Let's take the simplest case of using just the percentiles to compute the confidence interval. In that case you repeatedly sample with replacement from your data, compute your statistic in each of these samples and store those estimates. The 2.5th percentile of those stored estimates represents the lower bound and the 97.5th percentile the upper bound of a 95% confidence interval. 
If you have 200 replications that lower bound will be based on the 5th smallest estimate and the highest on the 5th highest. That will be way too small for my taste. My default is 20,000 replications, so the lower bound would be based on the 500th smallest estimate and the upper bound on the 500th largest estimate. The default is nothing but a starting point, and I will often choose another number depending on the exact circumstances.
A: According to Efron (the "inventor" of the boostrap technique), you should make 1600 replicas. I have no other clue about where this number comes from, except that its square root is 40, an easy number to divide by. I suggest you go like in any other Monte-Carlo. Try 1600, then increase the bootstrap samples until it stabilizes.
The bootstrap was introduced to compute confidence intervals in case the distribution of the v.a.r is unknown or not technically computable, because of outliers or skew. The bootstrap replaces the theoretical computations of the confidence interval by a measure of simulated samples. So the confidence intervals should be the same.
Note however that all statistical indicators are not equal in front of he boostrap. A average (or a mean) will require less samples than a maximum, or a 1% percentile for example.
