Statistica Sinica 27 (2017), 951-981
Abstract: Should one rely on a parametric or nonparametric model when analysing a given data set? This classic question cannot be answered by traditional model selection criteria like AIC and BIC, since a nonparametric model has no likelihood. The purpose of the present paper is to develop a focused information criterion (FIC) for comparing general non-nested parametric models with a nonparamet- ric alternative. It relies in part on the notion of a focus parameter, a population quantity of particular interest in the statistical analysis. The FIC compares and ranks candidate models based on estimated precision of the different model-based estimators for the focus parameter. It has earlier been developed for several classes of problems, but mainly involving parametric models. The new FIC, including also nonparametrics, is novel also in the mathematical context, being derived without the local neighbourhood asymptotics underlying previous versions of FIC. Cer- tain average-weighted versions, called AFIC, allowing several focus parameters to be considered simultaneously, are also developed. We concentrate on the standard i.i.d. setting and certain direct extensions thereof, but also sketch further generalisa- tions to other types of data. Theoretical and simulation-based results demonstrate desirable properties and satisfactory performance.
Key words and phrases: Asymptotic theory, focused information criterion, model selection.