INTRODUCTION TO BOOTSTRAP ESTIMATION
Bootstrapping is a general, distribution-free method that is used to estimate parameters of interest from data collected from studies or experiments. It is often referred to as a resampling method because it is carried out by repeatedly drawing samples from the original data that were gathered. This section introduces the basics of bootstrapping and extends it to bootstrapping in regression analysis. For a discussion on calculating bias or calculating confidence intervals using bootstrapping, see Efron and Tibshirani (1993).
Bootstrapping is a useful estimation technique when:
1. The formulas that are to be used for calculating estimates are based on assumptions that may not hold or may not be understood well, or cannot be verified, or are simply dubious.
2. The computational formulas hold only for large samples and are unreliable for small samples or simply not valid for small samples.
3. The computational formulas do not exist.
To begin the discussion of bootstrapping techniques, assume that a study or experiment was conducted resulting in a data set x1,…,xn of size n. This is a trivial case where the data are univariate in nature. Most studies involve collection of data on several variables as in the case of regression analysis studies. However, we use the simple example to lay the groundwork for the elements of bootstrapping methods.
Assume that the data set was generated by some underlying distribution f(θ). ...