Testing the fit of a parametric function

Abstract

General methods for testing the fit of a parametric function are proposed. The idea underlying each method is to ``accept'' the prescribed parametric model if and only if it is chosen by a model selection criterion. Several different selection criteria are considered, including one based on a modified version of the Akaike information criterion and others based on various score statistics. The tests have a connection with nonparametric smoothing because they use orthogonal series estimators to detect departures from a parametric model. An important aspect of the tests is that they can be applied in a wide variety of settings, including generalized linear models, spectral analysis, the goodness-of-fit problem, and longitudinal data analysis. Implementation using standard statistical software is straightforward. Asymptotic distribution theory for several test statistics is described, and the tests are shown to be consistent against essentially any alternative hypothesis. Simulations and a data example illustrate the usefulness of the tests.