Confidence Bands for a Distribution Function Using the Bootstrap

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Abstract
We first discuss the construction of bootstrap confidence bands for the distribution function F of a population for simple random sampling but do not assume that F is continuous. The known alternative approach is to use the quantiles from the tabled Kolmogorov distribution. This approach is known to be conservative. It has already been shown in Bickel and Freedman (1981) that the bootstrap confidence band has the correct coverage probability asymptotically. We show by simulation that the bootstrap works well for small samples and outperforms the conservative approach, particularly for distributions that have small carriers. We also investigate the analogous problem of finding bootstrap confidence bands for the distribution function F of a population for a more complicated situation, stratified random sampling. The conservative approach in the previous situation is extended to this case when sampling is with replacement (we expect that it holds for sampling without replacement) and the supports of the conditional distributions in each stratum are not overlapping. If the strata overlap there seems to be little alternative to the bootstrap. Use of the bootstrap in setting confidence bands or curves in this way should prove widely applicable, particularly when we leave the simple random sampling context as we have done. Asymptotic theory for the bootstrap confidence band is presented, and the conservative and bootstrap approaches are compared for small samples by simulation.

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