Shibata (1981, Biometrika 68, 45–54) considers
data-generating mechanisms belonging to a certain class of linear
regressions with errors that are independent and identically
normally distributed. He compares the variable selection criteria
AIC (Akaike information criterion) and BIC (Bayesian information
criterion) using the following type of comparison. For each
fixed possible data–generating mechanism, these criteria
are compared as the data length increases. The results of this
comparison have been interpreted as meaning that, in the context
of the data-generating mechanisms considered by Shibata, AIC
is better than BIC for large data lengths. Shibata's
comparison is pointwise in the space of data–generating
mechanisms (as the data length increases). Such comparisons
are potentially misleading. We consider a simple class of
data-generating mechanisms satisfying Shibata's assumptions
and carry out a different type of comparison. For each fixed
data length (possibly large) we compare the variable selection
criteria for every possible data-generating mechanism in this
class. According to this comparison, for this class of
data-generating mechanisms no matter how large the data length
AIC is not better than BIC.
Consistent variable selection criteria in multivariate linear regression even when dimension exceeds sample size
