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Stat 5601 (Geyer) Examples (Bootstrap Percentile)

Contents

• General Instructions
• Bootstrap Percentile Intervals
• Bootstrap Percentile Intervals, Take Two

General Instructions

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 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.

• We only look at all these quantiles because Efron and Tibshirani did. Usually, you only want two. For example,

  quantile(theta.star, probs = c(0.025, 0.975))
  

  for a 95% bootstrap percentile confidence interval or

  quantile(theta.star, probs = c(0.05, 0.95))
  

  for a 90% bootstrap percentile confidence interval.

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

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

In particular, if nboot is 999, then we take the ordered theta.star values to be the 0.001, 0.002, . . ., 0.999 quantiles of the sampling distribution of theta.hat. Thus

Comments

• If we wanted the quantiles to print with probabilities attached we would have to do

  fred <- sort(theta.star)[(nboot + 1) * probs]
  names(fred) <- probs
  fred
  

• But, of course, we don't want to do that unless we are writing a textbook like Efron and Tibshirani. Usually we just want the endpoints of a interval, either

  sort(theta.star)[(nboot + 1) * c(0.025, 0.975)]
  

  for a 95% bootstrap percentile confidence interval or

  sort(theta.star)[(nboot + 1) * c(0.05, 0.95)]
  

  for a 90% bootstrap percentile confidence interval.