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We study the switch distribution, introduced by Van Erven et al. (2012),applied to model selection and subsequent estimation. While switching was knownto be strongly consistent, here we show that it achieves minimax optimalparametric risk rates up to a $\log\log n$ factor when comparing two nestedexponential families, partially confirming a conjecture by Lauritzen (2012) andCavanaugh (2012) that switching behaves asymptotically like the Hannan-Quinncriterion. Moreover, like Bayes factor model selection but unlike standardsignificance testing, when one of the models represents a simple hypothesis,the switch criterion defines a robust null hypothesis test, meaning that itsType-I error probability can be bounded irrespective of the stopping rule.Hence, switching is consistent, insensitive to optional stopping and almostminimax risk optimal, showing that, Yang's (2005) impossibility resultnotwithstanding, it is possible to `almost' combine the strengths of AIC andBayes factor model selection.

Almost the Best of Three Worlds: Risk, Consistency and Optional Stopping for the Switch Criterion in Nested Model Selection