We study the switch distribution, introduced by Van Erven et al. (2012),
applied to model selection and subsequent estimation. While switching was known
to be strongly consistent, here we show that it achieves minimax optimal
parametric risk rates up to a loglogn factor when comparing two nested
exponential families, partially confirming a conjecture by Lauritzen (2012) and
Cavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinn
criterion. Moreover, like Bayes factor model selection but unlike standard
significance testing, when one of the models represents a simple hypothesis,
the switch criterion defines a robust null hypothesis test, meaning that its
Type-I error probability can be bounded irrespective of the stopping rule.
Hence, switching is consistent, insensitive to optional stopping and almost
minimax risk optimal, showing that, Yang's (2005) impossibility result
notwithstanding, it is possible to `almost' combine the strengths of AIC and
Bayes factor model selection.Comment: To appear in Statistica Sinic