Statistics 5601 (Geyer, Fall 2007) Bootstrap T

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.

Theory

All of the bootstrap t confidence intervals on this page are second order correct (Efron and Tibshirani, Chapter 22, p. 325).

Bootstrap T with Variance Estimate Supplied

Try 1

Section 12.5 in Efron and Tibshirani.

Comments

• As usual, library(bootstrap) says we are going to use code in the bootstrap library, which is not available without this command. Here library(bootstrap) is necessary for two reasons. Without it we can't get the data mouse.c and we also can't get the function boott (on-line help) that we use to construct bootstrap t intervals.

• The argument theta is a function that calculates the point estimate on which the interval is based. Here the point estimate is the sample mean, calculated by the function mean.

• The function sdfun calculates an estimate of the standard error of the estimator calculated by theta. In this example the sample mean (calculated by the mean function) has standard error calculated by sdfun.

• The call signature for sdfun is specified in the documentation for boott. It must have exactly this form, with three specified arguments and the dot, dot, dot argument.

• Generally, we only use the first argument x in our definition of sdfun. We can also use some of the others, but they are not so useful.
  • nbootsd is a dummy argument that is not used.
  • theta is the function that we passed in via the theta argument of boott (in this example, mean). We generally do not need to use the theta argument though, because we know the function: we can say mean(x) instead of theta(x).
  • ... indicates that other named arguments to the function are allowed. But we can also use global variables rather than arguments.

• The nboott = 1000 is because of the of the comment in the documentation for boott

  200 is a bare minimum and 1000 or more is needed for reliable α % confidence points, α > .95 say.

  and the similar comment near the top of p. 161 in Efron and Tibshirani.

Try 2

This is exactly the same as the preceding section but we make the output be only the confidence interval we want.

Bootstrap T with Double Bootstrap Variance Estimate

Section 12.5 in Efron and Tibshirani.

Comments

• If we omit the sdfun argument, then the standardization is done another way -- via the bootstrap. In each outer bootstrap iteration a sample x star with replacement from the data is formed, the function theta is applied to it to get theta star, the the standard error is estimated from x star by an inner bootstrap that forms samples x star star with replacement from x star and applies theta to them to get a bootstrap standard error.

  This function with sdfun omitted behaves as if we had supplied an sdfun of the following form

  sdfun <- function(x, nbootsd, theta, ...) {
      theta.star <- double(nbootsd)
      for (i in 1:nbootsd) {
          x.star <- sample(x, replace = TRUE)
          theta.star[i] <- theta(x.star)
      }
      return(sd(theta.star))
  }
  

  Note that when this sdfun is called with the original data x as an argument it just calculates the bootstrap standard error just like we did on our web page on that subject. But that when this sdfun is called with bootstrap data x.star as its argument x, then every x.star inside the function is a random sample without replacement from another random sample without replacement. Hence the name double bootstrap.

• As everywhere else, the default bootstrap sample sizes for these functions are too low. But cranking them up, as we do here, makes things really slow. In each of the nboott = 1000 iterations of the outer loop, the are nbootsd = 200 iterations of the inner loop, for 1000 * 200 = 2e5 iterations in all.

  If you get bored, you can use the defaults (omit nboott and nbootsd), but you won't get as accurate an answer.

Variance Stabilized Bootstrap T

Try 1

Section 12.6 in Efron and Tibshirani.

Another method of automagic variance stabilization estimates a variance stabilizing transformation using the double bootstrap.

Comments

• The function sally here calculates the estimator. It uses a trick from documentation for boott. It's a trick we have been using all along. For complicated data structures, sample the indices 1, . . ., n rather than the data vectors. So we just supply 1:n as the argument to boott and write the estimation function (here sally to take the (resampled) indices as an argument.

Try 2

This is exactly the same as the preceding section but we make the output be only the confidence interval we want, no plot showing the variance-stabilizing transformation.

Navigation

• University of Minnesota
• School of Statistics
• Rweb
• Stat 5601
  • Course Announcement
  • General Info
  • Homework Assignments
  • Homework Solutions
  • Exam Solutions
  • Computer Examples
  • Course Notes

Contents

• General Instructions
• Theory
• Bootstrap T with Variance Estimate Supplied
  • Try 1
  • Try 2
• Bootstrap T with Double Bootstrap Variance Estimate
• Variance Stabilized Bootstrap T
  • Try 1
  • Try 2