Confidence intervals centered on bootstrap smoothed estimators

TL;DR

This paper evaluates a confidence interval centered on bootstrap smoothed estimators in linear regression, demonstrating improved coverage probability and competitive length compared to traditional post-model-selection intervals.

Contribution

It derives practical formulas for the estimator and confidence interval performance, showing advantages over existing methods in nested linear models.

Findings

Confidence interval outperforms post-model-selection interval in coverage.

The interval has comparable expected length to the usual full-model interval.

Derived formulas facilitate efficient computation of coverage and length.

Abstract

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.