# Which bootstrap confidence intervals are provided by boot.ci in R?

I have reviewed a number of questions, but so far there is no clear answer on exactly which algorithms/procedures are implemented in boot.ci for R. For instance, 292619 mentions the types of intervals given but there it is in the context of selecting one of them, not defining the method.

The intervals are listed as:

• Normal
• Basic
• Studentized (Likely the bootstrap t, as in this paper, p. 1150)
• Percentile
• BCa (This one "appears" clearly defined!)

Unfortunately, these terms are not unambiguous as Michael Chernick points out, it is not clear if the percentile method is Efron's or Hall's (see comments on question). Additionally there is no clear linking from these methods to the sources. Nor do the docs is R clarify these points.

For instance, is the "normal" in this function the normal interval estimated by using the variance of the bootstrap distribution in the construction of the usual normal-theory CI? Or is that one the "basic" bootstrap? See here, page 216, if you can access this.

Is there a definitive source that lists the methods implemented and where the algorithms come from? Preferably a source that is available without trying to get a textbook?

• The documentation for boot.ci lists several references. One is a book. cran.r-project.org/web/packages/boot/boot.pdf Mar 2, 2018 at 18:24
• @Great38 -- That link only goes the the "man page" for the package and does not give any additional information. Also, there is no clear link to the citations given in the man page and the names of the methods. The Davison and Hinkley book is unavailable to me. :-( And reading the paper from Statistical Science there are multiple methods with the same/similar names mentioned, so linking them to the boot.ci output is not possible (for me anyway). Mar 2, 2018 at 18:34

From their help and code they have:

Normal: Using the normal approximation to a statistic, calculate equi-tailed two-sided confidence intervals.

Basic: Normal approximation method for confidence intervals. This can be used with or without a bootstrap object. If a bootstrap object is given then the intervals are bias corrected and the bootstrap variance estimate can be used if none is supplied.

Studentized: Studentized version of the basic bootstrap confidence interval.

Percentile: Bootstrap Percentile Confidence Interval Method. (they normalize it by Interpolation on the normal quantile scale. For a non-integer order statistic this function interpolates between the surrounding order statistics using the normal quantile scale. See equation 5.8 of Davison and Hinkley (1997)

I know that it's not much but I hope it helps a tiny bit.

• Do you interpret the "normal" as the interval constructed by estimating the standard error of the statistic from the bootstrap distribution? Or do you think it uses some bias correction? When I tested it it looked like the latter, but I was not sure. I agree that this is what the manual says, but unfortunately this is still vague. However your answer did lead me to the source code, and I am hoping that will help. Thanks for the input! Mar 2, 2018 at 20:26
• @Doctorambient if the bootstrap object is provided, then it uses bias correction. At least according to their code. Don't forget to upvote if the comment helped :). Mar 3, 2018 at 19:06