An Introduction to Bootstrap Methods with Applications to R

3

CONFIDENCE INTERVALS

Before we introduce the various bootstrap-type confidence intervals, we will review what confidence intervals are. In Section 3.1, we describe Efron's percentile method bootstrap and present Hartigan's typical value theorem as the motivation for Efron to introduce his version of the percentile method.

In general, what is the definition of a confidence region in two or more dimensions? Suppose we have a parameter vector ν that belongs in an n-dimension Euclidean space denoted by Rⁿ. A confidence region with confidence coefficient 1 − α is a set of points determined on the basis of a random sample and having the property that if the random sampling were repeated infinitely many times 100(1 − α)% of the generated set of points representing the confidence region will contain the true parameter and α% will not. In the one-dimensional case, the region will be an interval or a disjoint union of intervals if it is two sided and a region of the form (a, ∞) or (−∞, b) for one-sided regions.

In parametric models where nuisance parameters (parameters required to uniquely define the probability distribution for the sample observations but that are not of interest to the investigator) are involved or in nonparametric situations, it may not be possible to construct confidence regions for ν that have exactly the confidence level 1 − α for all possible values of the nuisance parameter(s) and all possible values of ν. The simplest common example of a nuisance parameter is ...