Confidence intervals centred on bootstrap smoothed estimators

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.KEYWORDS confidence interval; coverage probability; decision theory; bootstrap smoothed estimator; model selection; nested linear regression models; scaled expected length.

Parsimonious model selection using information theory: a modified selection rule

“…This is an active area of research (Claeskens and Hjort 2008, Berk et al. 2013, Charkhi and Claeskens 2018, Kabaila and Wijethunga 2019). For predictive inference, the post‐selection confidence interval of the mean response can be estimated using techniques based on model averaging (Hjort and Claeskens 2003, Efron 2014).…”

Model averaged tail area confidence intervals in nested linear regression models

“…He then considered a confidence interval, with nominal coverage $$$ 1-\alpha $$$, centred on this estimator and with half‐width equal to the $$$ 1-\alpha /2 $$$ quantile of a standard normal distribution multiplied by an estimate of this delta method approximation. The performances, in terms of coverage and expected length, of this confidence interval and the frequentist model averaged confidence interval of Buckland, Burnham & Augustin (1997), together with the variants of Burnham & Anderson (2002) and Lukacs, Burnham & Anderson (2010), were assessed by (Kabaila & Wijethunga 2019a; Kabaila & Wijethunga 2019b) and Kabaila, Welsh & Wijethunga (2020), using a testbed consisting of two nested linear regression models. These assessments illustrate the difficulty of finding confidence intervals that perform better than the standard confidence interval obtained by fitting the full model.…”