Parametric or nonparametric: The FIC approach

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 nonparametric 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. Certain 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 generalisations to other types of data. Theoretical and simulation-based results demonstrate desirable properties and satisfactory performance.

Focused information criteria for copulas

“…This paper concerns extending the FIC theory to these different estimation procedures. This partly follows the line of arguments used in Jullum and Hjort (, ) but, by necessity, involves certain extra efforts, as we work with more complex estimation strategies than the pure nonparametric and the ML.…”

“…In this subsection, we briefly summarize the idea behind the FIC of Jullum and Hjort () and reach a general formula. The actual details for implementing this FIC formula depend on the specific estimation scheme, which we return to later in Sections 2.2, 2.3, and 2.4.…”

“…However, in many cases, the model itself is not the goal, but rather a means to estimate a specific quantity, such as the mean or the probability of a certain event. The main idea behind the FIC, first introduced by Claeskens and Hjort () and later modified by Jullum and Hjort (, ), is that one wants to select a model that is good for a specific task. In our setting, we quantify this specific task as that of estimating a focus parameter, say, T ( G ), a functional of the data‐generating distribution.…”

“…We also need the functional to be smooth, in the sense of admitting an influence function leading to limiting normality with a rate. This involves certain regularity conditions, as (C1)–(C4) of Jullum and Hjort (, Section 2).…”

“…This quantity of interest is called the “focus parameter” and hence the name “focused information criterion.” One of the drawbacks of the original FIC (Claeskens & Hjort, ) is that it assumes all the models to be parametric and nested. Jullum and Hjort () extend the FIC to a more general setting by using a nonparametric model to estimate the bias part of the decomposed MSE. Their new FIC machinery is designed for maximum likelihood (ML)‐estimated parametric models and nonparametric models.…”