An Introduction to Bootstrap Methods with Applications to R

2

ESTIMATION

This chapter covers point estimation. Section 2.1 deals with ways the bootstrap can be used to estimate the bias of an estimate and “improve” it by a bias adjustment. Historically, the bootstrap was looked at to estimate the standard error of an estimate and later on for bias adjustment, while its relative the jackknife was first used for bias adjustment and then later it found a place in estimating standard errors. Section 2.1 covers bias adjustment in general, and then the application to error rate estimation in discriminant analysis is presented. Sections 2.2 and 2.3 cover measures of location and spread, respectively, as parameters to estimate by bootstrapping. Then, we look at linear regression parameters in Section 2.4 and nonlinear regression parameters in Section 2.5. Section 2.7 is the section on historical notes, and it is followed by exercises (Section 2.8) and references.

2.1 ESTIMATING BIAS

2.1.1 Bootstrap Adjustment

Let us denote the expectation of a random variable X by E(X). Let be an estimate of a parameter θ. Then, consider the quantity to be the random variable X. The bias for the estimator of θ is .

EXAMPLE 2.1

As an example, let us consider for univariate ...