4) note that HQC, like BIC, but unlike AIC, is not asymptotically efficient; however, it misses the optimal estimation rate by a very small
They further point out that whatever method is being used for fine-tuning the criterion will be more important in practice than the term
term ensures that, unlike AIC, HQC is strongly consistent.
It follows from the law of the iterated logarithm that any strongly consistent method must miss efficiency by at least a
Van der Pas and Grünwald prove that model selection based on a modified Bayesian estimator, the so-called switch distribution, in many cases behaves asymptotically like HQC, while retaining the advantages of Bayesian methods such as the use of priors etc.