Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. The key result of Efron (2014) is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron (2014) proposed a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard deviation. We evaluate this confidence interval in the scenario of two nested linear regression models, the full model and a simpler model, and a preliminary test of the null hypothesis that the simpler model is correct. We derive computationally convenient expressions for the ideal bootstrap smoothed estimator and the coverage probability and expected length of this confidence interval. In terms of coverage probability, this confidence interval outperforms the post-model-selection confidence interval with the same nominal coverage and based on the same preliminary test. We also compare the performance of confidence interval centered on the bootstrap smoothed estimator, in terms of expected length, to the usual confidence interval, with the same minimum coverage probablility, based on the full model.

中文翻译：

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以自举平滑估计量为中心的置信区间

Bootstrap 平滑（袋装）参数估计器已被提议作为对基于初步数据的模型选择后发现的估计器的改进。Efron (2014) 的主要结果是一个非常方便且广泛适用的公式，用于近似自举平滑估计器标准差的 delta 方法。该近似值为该估计器的准确性提供了一个易于计算的指南。此外，Efron (2014) 提出了一个以 bootstrap 平滑估计器为中心的置信区间，其宽度与该标准差的近似值的估计值成正比。我们在两个嵌套线性回归模型（完整模型和更简单模型）的场景中评估此置信区间，并初步检验了更简单模型正确的零假设。我们推导出了理想自举平滑估计器的计算方便表达式以及该置信区间的覆盖概率和预期长度。在覆盖概率方面，此置信区间优于具有相同标称覆盖率和基于相同初步测试的模型选择后置信区间。我们还比较了以自举平滑估计器为中心的置信区间的性能，在预期长度方面，与基于完整模型的具有相同最小覆盖概率的通常置信区间。该置信区间优于模型选择后置信区间，具有相同的标称覆盖率并基于相同的初步测试。我们还比较了以自举平滑估计器为中心的置信区间的性能，在预期长度方面，与基于完整模型的具有相同最小覆盖概率的通常置信区间。该置信区间优于模型选择后置信区间，具有相同的标称覆盖率并基于相同的初步测试。我们还比较了以自举平滑估计器为中心的置信区间在预期长度方面的性能，与基于完整模型的具有相同最小覆盖概率的通常置信区间的性能。