  In model selection literature, two classes of criteria perform well
asymptotically in different situations: Bayesian information criterion (BIC)
(as a representative) is consistent in selection when the true model is finite
dimensional (parametric scenario); Akaike's information criterion (AIC)
performs well in an asymptotic efficiency when the true model is infinite
dimensional (nonparametric scenario). But there is little work that addresses
if it is possible and how to detect the situation that a specific model
selection problem is in. In this work, we differentiate the two scenarios
theoretically under some conditions. We develop a measure, parametricness index
(PI), to assess whether a model selected by a potentially consistent procedure
can be practically treated as the true model, which also hints on AIC or BIC is
better suited for the data for the goal of estimating the regression function.
A consequence is that by switching between AIC and BIC based on the PI, the
resulting regression estimator is simultaneously asymptotically efficient for
both parametric and nonparametric scenarios. In addition, we systematically
investigate the behaviors of PI in simulation and real data and show its
usefulness.
