Statistics 5601 (Geyer, Spring 2006) Examples: Bootstrap Percentile

General Instructions
Bootstrap Percentile Intervals
Bootstrap Percentile Intervals, Take Two

General Instructions

To do each example, just click the "Submit" button. You do not have to type in any R instructions or specify a dataset. That's already done for you.

Bootstrap Percentile Intervals

Section 13.3 in Efron and Tibshirani.

Comments

Everything down to the bottom of the for loop should be familiar, just like what we did calculating bootstrap standard errors (sd(theta.star) would be the bootstrap standard error).
The R function quantile (on-line help) calculates quantiles of a data vector, at least, what it calls quantiles. Its definition is a bit eccentric, but is asymptotically equivalent to all other definitions of quantiles.
If you want, for example, a 90% equal-tailed confidence interval, you replace the definition of conf.level by
```
conf.level <- 0.90
```

Bootstrap Percentile Intervals, Take Two

An alternative method for quantiles preferred by your humble instructor uses the following logic.

Use nboot <- 999 (or some other value such that nboot + 1 is a round number. The reason is that if X_(i) is the i-th order statistic from a Uniform(0, 1) distribution

E{X_(i)} = i / (n + 1)

Another way to think of this is that the nboot data points divide the number line into nboot + 1 intervals, which as far as we know contain equal probability. They don't contain equal probability because the sample is not the population, but we might as well treat them as such for the purposes of estimation. That is, our nboot data points should be taken as estimators of the quantiles with denominators nboot + 1