Confidence Interval from bootstrap

To produce the , say, 95%, confidence interval(CI) from the bootstrap distribution, I know 2 approaches:

Approach 1: calculate the 2.5% and 97.5% percentile from the bootstrap distribution

Approach 2: bootstrap mean +/- 1.96*bootstrap SE

I would like to ask in which cases each approach will be more sensible and why so.

There are at least 4 or 5 types of Bootstrap confidence intervals (BCa, bootstrap-t, ABC, and calibration). These are thoroughly described here:

Bootstrap Confidence Intervals. Thomas J. DiCiccio and Bradley Efron

And they are all implemented in the 'boot' R package.

Some ideas I have read about in Rand R. Wilcox - Fundamentals of Modern Statistical Methods, which by the way, it is a really nice book to read in general.

Approach 1 which is called percentile bootstrap works well only if you have a pretty number of observations and it covers well the whole interval of interest.

Approach 2 which is called percentile t bootstrap is slightly better for smaller samples than the previous samples. However it is interesting to check if the distribution resembles a normal distribution. If this is the case a t statistic is better than percentile t bootstrap. The bootstrap provides advantages only when the distribution is not normal.

I always used the second method. However what I do usually is to employ both methods and if they provides very different results than try to identify why that happened and chose the variant which makes those assumptions which fits better my data.

• @Tim I don't think so. The sample mean distribution is normal by CLT when sample size goes to infinity, no matter how the original distribution is. Now the sample size does not go to infinity and also the distribution is not always friendly. Taking bootstraps alleviates that, but notice that the sample mean of a bootstrap still has a normal distribution under same assumptions via CLT. Considering that you can use t distribution since you estimate also variance from the same sample. – rapaio Nov 17 '16 at 10:03
• thanks a lot for your clear answer, it definitely helps. I'll look into the book you recommended. – Long Ngo Nov 17 '16 at 15:37