This has two profound implications.
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First, it means that we do not use the mean of the bootstrap statistics as a replacement for the original estimate. Instead we use the bootstrap to tell how accurate the original estimate is. In this regard the bootstrap is like formula methods that use the data twice—once to compute an estimate, and again to compute a standard error for the estimate. The bootstrap just uses a different approach to estimating the standard error.
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We generally do not do this—bias estimates can have high variability Efron and Tibshirani What people may think of as the key bootstrap idea—drawing samples with replacement from the data—is just a pair of implementation details. The first is substituting the empirical distribution for the population; alternatives include smoothed or parametric distributions.
The second is using random sampling. Here too there are alternatives, including analytical methods e. There are n n possible bootstrap samples from a fixed sample of size n , 2 n - 1 n if order does not matter, or even fewer in some cases like binary data; if n is small we could evaluate all of these. We call this an exhaustive bootstrap or theoretical bootstrap. But more often exhaustive methods are infeasible, so we draw say 10, random samples instead; we call this the Monte Carlo sampling implementation. Normally we should draw bootstrap samples the same way the sample was drawn in real life, for example, simple random sampling or stratified sampling.
Pedagogically, this reinforces the role that random sampling plays in statistics. One exception to that rule is to condition on the observed information.
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For example, when comparing samples of size n 1 and n 2 , we fix those numbers, even if the original sampling process could have produced different counts. This is the conditionality principle in statistics, the idea of conditioning on ancillary statistics. We can also modify the sampling to answer what-if questions. For example, we could bootstrap with and without stratification and compare the resulting standard errors, to investigate the value of stratification. We could also draw samples of a different size; say we are planning a large study and obtain an initial dataset of size , we can draw bootstrap samples of size to estimate how large standard errors would be with that sample size.
Conversely, this also answers a common question about bootstrapping—why we sample with the same size as the original data—because by doing so the standard errors reflect the actual data, rather than a hypothetical larger or smaller dataset. We claimed above that the bootstrap distribution usually provides useful information about the sampling distribution.
We elaborate on that now with a series of visual examples, one where things generally work well and three with problems. We address two questions:.
How accurately does the Monte Carlo implementation approximate the theoretical bootstrap? The spreads and shapes of the bootstrap distributions vary a bit but not a lot. These observations inform what the bootstrap distributions may be used for. Similarly, quantiles of the bootstrap distributions are not useful for estimating quantiles of the sampling distribution. Instead, the bootstrap distributions are useful for estimating the spread and shape of the sampling distribution.
Using more resamples reduces random Monte Carlo variation, but does not fundamentally change the bootstrap distribution—it still has the same approximate center, spread, and shape.
The Monte Carlo variation is much smaller than the variation due to different original samples. As before, the bootstrap distributions are centered at the corresponding sample means, but now the spreads and shapes of the bootstrap distributions vary substantially, because the spreads and shapes of the samples vary substantially. As a result, bootstrap confidence interval widths vary substantially this is also true of standard t confidence intervals. As before, the Monte Carlo variation is small and may be reduced with more resamples.
Bootstrap percentile intervals lack these factors, so tend to be too narrow and under-cover in small samples.
Some other bootstrap procedures do better. For Stat I suggest warning students about the issue; for higher courses you may discuss remedies Hesterberg , In two-sample or stratified sampling situations, the narrowness bias depends on the individual sample or strata sizes. This can result in severe bias. For example, the U. Department of Work and Pensions wanted to bootstrap a survey of welfare cheating.
Here the bootstrap distributions are poor approximations of the sampling distribution. The sampling distribution is continuous, but the bootstrap distributions are discrete—for odd n the bootstrap sample median is always one of the original observations—and with wildly varying shapes. The ordinary bootstrap tends not to work well for statistics such as the median or other quantiles in small samples that depend heavily on a small number of observations out of a larger sample.
The bootstrap depends on the sample accurately reflecting what matters about the population, and those few observations cannot do that. The right column shows the smoothed bootstrap ; it is better, though is still poor for this small n. In spite of the inaccurate shape and spread of the bootstrap distributions, the bootstrap percentile interval for the median is not bad Efron For odd n , percentile interval endpoints fall on one of the observed values.
In many applications, the spread or shape of the sampling distribution depends on the parameter of interest. For example, the binomial distribution spread and shape depend on p. This mean—variance relationship is reflected in bootstrap distributions. This relationship has important implications for confidence intervals; procedures that ignore the relationship are inaccurate. There are other applications where sampling distributions depend strongly on the parameter, for example, sampling distributions for chi-squared statistics depend on the noncentrality parameter.
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Use caution when bootstrapping such applications; the bootstrap distribution may be very different from the sampling distribution. Here there is a bright spot. These distributions are much less sensitive to the original sample. The bootstrap distribution reflects the original sample. If the sample is narrower than the population, the bootstrap distribution is narrower than the sampling distribution.
Typically for large samples the data represent the population well; for small samples they may not.
Bootstrapping does not overcome the weakness of small samples as a basis for inference. Indeed, for the very smallest samples, it may be better to make additional assumptions such as a parametric family. Another visual lesson is that random sampling using only resamples causes more random variation in the bootstrap distributions.
Let us consider this issue more carefully. I suggested above using bootstrap samples for rough approximations, or 10 4 or more for better accuracy.
This is about Monte Carlo accuracy—how well the usual Monte Carlo implementation of the bootstrap approximates the theoretical bootstrap distribution. A bootstrap distribution based on r random samples corresponds to drawing r observations with replacement from the theoretical bootstrap distribution.
I argue that more resamples are appropriate. First, computers are faster now. Second, those criteria were developed using arguments that combine variation due to the original random sample with the extra variation from the Monte Carlo implementation. I prefer to treat the data as given and look just at the variability due to the implementation.
Two people analyzing the same data should not get substantially different answers due to Monte Carlo variation. We can quantify the Monte Carlo variation in two ways—using formulas, or by bootstrapping. We can also bootstrap the bootstrap distribution! The r bootstrap statistics are an iid sample from the exhaustive bootstrap distribution; we can bootstrap that sample.
To estimate the accuracy of those quantiles, we draw resamples of size r from the bootstrap distribution and compute the quantiles for each resample. The resulting SEs for the quantile estimates are 0.
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