Confidence Bands for a Distribution Function Using the Bootstrap
使用 Bootstrap 构造分布函数置信带AI 解读 ↓
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使用 Bootstrap 构造分布函数置信带
- 研究问题
- 在总体分布可能不连续、抽样可能分层时，如何构造分布函数置信带，并改善传统保守方法的小样本表现？
- 方法与思路
- 作者研究简单随机抽样与分层抽样下的 Bootstrap 置信带，讨论渐近理论，并通过模拟与保守方法比较。
- 材料陈述的贡献
- 摘要称，Bootstrap 在所用小样本模拟中表现良好，尤其适用于支持点较少的分布；文章还考察更复杂的分层情形。
- 阅读关注点
- 应核对有放回与无放回抽样的区别、各层支持集重叠的影响，以及渐近覆盖结论与小样本模拟的对应关系。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明，不能推断刊本变化；对无放回情形的部分表述只是作者预期，不能视为已证明结果。
原文摘要
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.