Parametric or nonparametric? A parametricness index for model selection

TL;DR

This paper introduces a parametricness index (PI) to determine whether a model selection problem is parametric or nonparametric, guiding the choice between BIC and AIC for optimal regression estimation.

Contribution

The paper develops a theoretical measure, PI, to identify the scenario and adapt model selection criteria accordingly, improving estimation efficiency.

Findings

PI effectively differentiates parametric and nonparametric scenarios

Switching between AIC and BIC based on PI improves estimation accuracy

PI demonstrates practical usefulness in simulations and real data

Abstract

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.