Bootstrap Confidence Intervals

There are several ways to construct bootstrap confidence intervals

Normal Interval

The Normal Interval

The simplest method is the Normal interval

where is the bootstrap estimate of the standard error.

Warning

This interval is not accurate unless the distribution of is close to Normal.

Pivotal Intervals

Let and and define the pivot . Let denote the bootstrap replications of . Let denote the CDF of the pivot:

Define where

It follows that

Hence, is an exact confidence interval for . Unfortunately, and depend on the unknown distribution of , but we can form a bootstrap estimate of :

where . Let denote the sample quantile of and let denote the sample quantile of . Note that . It follows that an approximate confidence interval is where

Bootstrap Pivotal Confidence Interval

The bootstrap pivotal confience interval is

Intuition

The pivot is the estimation error. If we knew the distribution of this error, then we could turn typical errors into a confidence interval for by solving for the values of that make the observed estimate look typical.

The bootstrap mimics this unknown error distribution by replacing the unknown truth with the observed estimate . In the bootstrap world, the analogue of the error is

So the bootstrap pivotal interval says: look at the typical bootstrap errors, then reflect them around to get plausible values of the true parameter. This is why the endpoints have the form

Equivalently, if the bootstrap estimates are unusually far above , then plausible values of should be correspondingly far below , since those values would make the observed estimate have the same size error.

Theorem

Under weak conditions on ,

as , where is given above.

Sources

• Wasserman, L. (2010). All of Statistics: A concise Course in Statistical Inference. Chapter 8.3.