Statistics 5601 (Geyer, Fall 2013) Subsampling

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.

Overview

The subject of this web page is the subsampling bootstrap, which is the subject of a book by Politis, Romano, and Wolfe.

It is also the subject of a more detailed web page, which we will get to in a few weeks.

Basic Idea

The subsampling bootstrap samples without replacement at a subsample size b that is smaller than the original sample size n. The sampling without replacement has the consequence that the samples are from the true unknown population distribution.

• The Efron nonparametric bootstrap samples from the wrong distribution (the empirical) at the right sample size n.
• The Politis and Romano subsampling (nonparametric) bootstrap samples from the right distribution at the wrong sample size b.

Neither does both things right. Each has its virtues. For the subsampling bootstrap we need b large but small compared to n, so n must be really large.

Rate of Convergence

In order to use the subsampling bootstrap we must know the rate of convergence of the estimator we are using. We assume that if t_n is the estimator, θ is the parameter, and n is the sample size, then

converges in distribution (to some, not necessarily normal, distribution).

We estimate this distribution by the distribution of

where b is the subsample size and t_b* is the subsampling bootstrap estimator.

Often r is 1 ⁄ 2 (the square root law obeyed by most widely used estimators). Sometimes, as in the extreme values example below, it is not.

Stationary Process or IID Sampling

There are two ways to do subsampling.

One is essential for stationary time series and is demonstrated in the time series example below. In this method, the subsamples are all blocks of length b in the time series. There are not many such blocks (n − b + 1), but it is necessary to keep the blocks together to keep the dependence in the time series (at least the dependence that is present in blocks of length b).

The other method applies only to IID and is demonstrated in the extreme values example below. In this method, the subsamples are samples without replacement of length b from the original sample. This allows many more samples than the other method and a more accurate bootstrap.

Time Series

Sections 8.5 and 8.6 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.

• The lutenhorm data is explained by its on-line help. Inspection of the lutenhorm dataset shows that column 4 is the data described in Table 8.1 in Efron and Tibshirani.

• The function acf on-line help calculates the so-called autocorrelation function of the time series. The height of the bar at lag k is the correlation of X_n and X_{n + k} assuming the time series is stationary (so this correlation does not depend on n only on k. The correlation at lag zero is one by definition (any random variable is perfectly correlated with itself).

  The blue dashed lines in the autocorrelation plot are 95% non-simultaneous large sample approximate critical values for testing whether the autocorrelations are non-zero. Autocorrelations that go outside the blue dashed lines are statistically significant. Here only the lag 1 autocorrelation is significant.

• The function foo calculates the estimator of the autoregressive coefficient described by Efron and Tibshirani.

  The vector z is the data supplied to the function with the mean subtracted off. The number m is the length of the data.

  The vector z[-1] is all the elements of z except the first and the vector z[-m] is all the elements of z except the last.

  Thus the statement

  out <- lm(z[-1] ~ z[-m] + 0)
  

  regresses z_t on z_{t − 1} with no intercept (the + 0 means no intercept).

• For time series, the subsampling bootstrap uses only blocks of contiguous variables. For a series of length n and blocks of b there are exactly n - b + 1 such blocks. Generally, we use them all. No need for random samples.

• Note that the bootstrap samples are correlated, as the time series plot for beta.star shows. However, this does not matter, so long as b is long enough so the samples are representative of the behavior of the whole series.

  As usual, Efron and Tibshirani are using a ridiculously small sample size in this toy problem. There is no reason to believe the subsampling bootstrap here. But it is reasonable for (much) larger data sets.

• The histogram of beta.star shows that the simple method of estimation being used here is badly biased. That's why this method is not recommended by time series books. We only use it here because it is easy to explain.

• The sqrt(b / n) in the last line adjusts for the relative sample sizes of the subsample and the whole series. Note that the sqrt here is only valid for estimators obeying the square root law. If the rate is not root n, then a different function of b / n is needed, as in the following example.

Extreme Values and IID in General

This section is about using the subsampling bootstrap in situations where the data are IID (independent and identically distributed) but in which the ordinary nonparametric bootstrap (Efron's nonparametric bootstrap) does not work. The only difference for a different problem is that the rate might be different. In this example we have rate n. In another problem we might have rate n^{1 / 3}. In yet another we might have, yet another rate.

Section 7.4 in Efron and Tibshirani.

Comments

• The vector theta.star stores max(x) for samples from the subsampling bootstrap.

• The vector theta.bogo stores max(x) for samples from the ordinary (Efron) bootstrap.

• Note that the sample statement is quite different for the regular (Efron) bootstrap and the (Politis and Romano) subsampling bootstrap.

  For the Efron bootstrap, we sample with replacement at the original sample size with something like

  x.star <- sample(x, replace = TRUE)
  

  The subsampling bootstrap samples without replacement at the much smaller sample size b with something like

  x.star <- sample(x, b, replace = FALSE)
  

  Both the size and the replace arguments of sample differ. (For the Efron bootstrap the size argument is missing so the default length(x) is used.)

• Since the asymptotic distribution is non-normal, it makes no sense to be calculating standard errors. What does make sense is a bootstrap percentile interval, but we will have to wait until we learn about that and revisit this issue.

• For now, we just show that the subsampling bootstrap has done the Right Thing (with a capital R and a capital T). The plot is a so-called Q-Q plot. The sorted values of the variable

  z.star <- b * (theta.hat - theta.star)
  

  which is supposed to have an Exp(1 / θ) distribution according to the theory, are plotted against the appropriate quantiles of this distribution. If the points lie near the line y = x, then z.star does indeed have the claimed distribution.

  We emphasize that we don't need to know the asymptotic distribution to use the bootstrap samples z.star to construct a confidence interval for θ. We can't do it yet because we haven't covered Chapters 12, 13, and 14 in Efron and Tibshirani. When we've done them, we can return to this example and finish it.

• For comparison, we put also the theta.bogo samples on the Q-Q plot, so it can be clearly seen they do the Wrong Thing (with a capital W and a capital T).

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Contents

• General Instructions
• Overview
• Time Series
• Extreme Values and IID in General