  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.
