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.


Now that we have some idea of how (Efron style) bootstrap confidence intervals work, we can return to the subsampling bootstrap and look at the kind of confidence intervals Politis and Romano (1994) recommend for the subsampling bootstrap.

Somewhat suprisingly, they recommend the kind of intervals that Efron and Tibshirani disparage in Section 13.4 titles Is the Percentile Interval Backwards?. This section is a response to the recommendations of Hall (The Bootstrap and Edgeworth Expansion, Springer, 1992, and earlier papers) where he charactizes bootstrap percentile intervals as looking up [in] the wrong statistical tables backwards.

We take no sides on the argument between Efron and Hall. There are arguments on both sides, and either method performs well in some examples and badly in others.

However, for the subsampling bootstrap, there seems to be no dispute. The standard method for the subsampling bootstrap is very much like what Hall recommends for the ordinary bootstrap.

This method is explained in a handout (which will be handed out in class). For those who didn't get a copy in dead tree format, here is a PDF.

Time Series


Time Series Revisited

In this section we look at a long AR(1) time series, so long that it is silly to use all blocks of length b as subsamples. We take a sample of the subsamples.

Also we use a sound method of estimating the AR coefficient.

The R function ar.burg (on-line help) estimates AR(k) time series models for arbitrary k and also determines k. Here we want the case k = 1, which makes the call a bit complicated (the default is to estimate k). Another complication is that this function returns a list, the ar element of which is the estimate. That's why the $ar at the end of the function call. It picks out that element of the list.

The command spc <- 100 says we will use every 100-th block of length b. This complicates the specification of the subsamples. The starting indices of the blocks we use are given by the vector ii and the i-th time through the loop the subsample will be the block starting at ii[i] and ending at ii[i] + b - 1. This is just like the code where we use all the subsamples except we have replaced i by ii[i] at this point.

Everything else is the same as the preceding example.

Extreme Values and IID in General


Estimating the Rate