Can the strengths of AIC and BIC be shared? A conflict between model indentification and regression estimation

Yuhong Yang

Research output: Contribution to journalArticlepeer-review

442 Scopus citations

Abstract

A traditional approach to statistical inference is to identify the true or best model first with little or no consideration of the specific goal of inference in the model identification stage. Can the pursuit of the true model also lead to optimal regression estimation? In model selection, it is well known that BIC is consistent in selecting the true model, and AIC is minimax-rate optimal for estimating the regression function. A recent promising direction is adaptive model selection, in which, in contrast to AIC and BIC, the penalty term is data-dependent. Some theoretical and empirical results have been obtained in support of adaptive model selection, but it is still not clear if it can really share the strengths of AIC and BIC. Model combining or averaging has attracted increasing attention as a means to overcome the model selection uncertainty. Can Bayesian model averaging be optimal for estimating the regression function in a minimax sense? We show that the answers to these questions are basically in the negative: for any model selection criterion to be consistent, it must behave suboptimally for estimating the regression function in terms of minimax rate of covergence; and Bayesian model averaging cannot be minimax-rate optimal for regression estimation.

Original languageEnglish (US)
Pages (from-to)937-950
Number of pages14
JournalBiometrika
Volume92
Issue number4
DOIs
StatePublished - Dec 2005

Bibliographical note

Funding Information:
The author thanks two anonymous referees and the editor for their helpful comments. This work was supported by a CAREER Grant from the U.S. National Science Foundation.

Keywords

  • AIC
  • BIC
  • Consistency
  • Minimax-rate optimality
  • Model averaging
  • Model selection

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