Testing the fit of a parametric function

General methods for testing the fit of a parametric function are proposed. The idea underlying each method is to "accept" the prescribed parametric mo del if and only if it is chosen by a model selection criterion. Several dif ferent selection criteria are considered, including one based on a modified version of the Akaike information criterion and others based on various sc ore statistics. The tests have a connection with nonparametric smoothing be cause they use orthogonal series estimators to detect departures from a par ametric model. An important aspect of the tests is that they can be applied in a wide variety of settings, including generalized linear models, spectr al analysis, the goodness-of-fit problem, and longitudinal data analysis, i mplementation using standard statistical software is straightforward. Asymp totic distribution theory for several test statistics is described, and the tests are shown to be consistent against essentially any alternative hypot hesis. Simulations and a data example illustrate the usefulness of the test s.