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Biometrika Advance Access published online on May 15, 2007
Biometrika, doi:10.1093/biomet/asm034
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Copyright © 2007 Biometrika Trust

Article

An asymptotic theory for model selection inference in general semiparametric problems

Gerda Claeskens

Operations Research & Business Statistics and University Center for Statistics, Katholieke Universiteit Leuven, Naamsestraat 69, B-3000 Leuven, Belgium

Raymond J. Carroll

Department of Statistics, Texas A&M University, College Station, Texas 77843-3143, U.S.A.

gerda.claeskens{at}econ.kuleuven.be

carroll{at}stat.tamu.edu

Received for publication 1 May 2005. Revision received 1 October 2006.
  Abstract

Hjort & Claeskens (2003) developed an asymptotic theory for model selection, model averaging and subsequent inference using likelihood methods in parametric models, along with associated confidence statements. In this article, we consider a semiparametric version of this problem, wherein the likelihood depends on parameters and an unknown function, and model selection/averaging is to be applied to the parametric parts of the model. We show that all the results of Hjort & Claeskens hold in the semiparametric context, if the Fisher information matrix for parametric models is replaced by the semiparametric information bound for semiparametric models, and if maximum likelihood estimators for parametric models are replaced by semiparametric efficient profile estimators. Our methods of proof employ Le Cam's contiguity lemmas, leading to transparent results. The results also describe the behaviour of semiparametric model estimators when the parametric component is misspecified, and also have implications for pointwise-consistent model selectors.

Key Words: Akaike information criterion • Bayes information criterion • Efficient semiparametric estimation • Frequentist model averaging • Model averaging • Model selection • Profile likelihood • Semiparametric model


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