PARAMETRIC OR NONPARAMETRIC? A PARAMETRICNESS INDEX FOR MODEL SELECTION

• 成果类型: Article
• 署名作者: Liu, Wei; Yang, Yuhong
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/11-AOS899
• 发表日期: 2011
• 页码: 2074-2102
• 关键词: VARIABLE SELECTION; ASYMPTOTIC THEORY; regression; density; inference; criteria; Consistency; complexity; prediction; likelihood

摘要

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.