Confidence intervals centred on bootstrap smoothed estimators: an impossibility result

TL;DR

This paper proves that it is impossible to construct a confidence interval centered on a bootstrap smoothed estimator that simultaneously maintains desired coverage and scaled expected length properties in a specific linear regression testing scenario.

Contribution

It demonstrates, using a decision-theoretic bound, that no data-based width formula can achieve optimal coverage and length properties for such confidence intervals in the studied setting.

Findings

No suitable data-based width formula exists for the confidence interval.

The scaled expected length can be unacceptably large or not sufficiently small.

Impossibility holds across a wide range of scenarios.

Abstract

Recently, Kabaila and Wijethunga assessed the performance of a confidence interval centred on a bootstrap smoothed estimator, with width proportional to an estimator of Efron's delta method approximation to the standard deviation of this estimator. They used a testbed situation consisting of two nested linear regression models, with error variance assumed known, and model selection using a preliminary hypothesis test. This assessment was in terms of coverage and scaled expected length, where the scaling is with respect to the expected length of the usual confidence interval with the same minimum coverage probability. They found that this confidence interval has scaled expected length that (a) has a maximum value that may be much greater than 1 and (b) is greater than a number slightly less than 1 when the simpler model is correct. We therefore ask the following question. For a confidence interval, centred on the bootstrap smoothed estimator, does there exist a formula for its data-based width such that, in this testbed situation, it has the desired minimum coverage and scaled expected length that (a) has a maximum value that is not too much larger than 1 and (b) is substantially less than 1 when the simpler model is correct? Using a recent decision-theoretic performance bound due to Kabaila and Kong, it is shown that the answer to this question is `no' for a wide range of scenarios.