Mathematical Statistics with Resampling and R

11

ADDITIONAL TOPICS

This chapter includes a variety of topics. We will begin by considering the fundamental bootstrap principle—plugging in an estimate for the population in place of the population and looking at two new possibilities for what to plug in—in Sections 11.1 (smoothed bootstrap) and 11.2 (parametric bootstrap). We will then consider some computational methods. One motivation is for use in Bayesian analysis in nonconjugate situations, but the methods are useful in general. We will discuss the delta method for finding standard errors, stratified sampling, Monte Carlo integration, and importance sampling.

11.1 SMOOTHED BOOTSTRAP

Recall that a sampling distribution is the distribution of a statistic when drawing random samples from a population. In practice, drawing thousands or millions of repeated samples from the population is impossible, so the fundamental bootstrap idea is to draw samples from an estimate of the population.

In earlier chapters, we sampled from the observed data, with empirical cumulative distribution function where #{x_i ≤ x} is the number of x_i values that are below x.

But that is not always a good choice. Recall Section 5.8, where we found that the ordinary bootstrap does not work well for the median; the bootstrap distributions in Figure 5.20 on page 128 look nothing like the sampling distribution.

Figure 11.1 shows another view of why that happened. ...