Unified approach to testing functional hypotheses in semiparametric contexts

We suggest a new method, with very wide applicability, for testing semiparametric hypotheses about functions such as regression means and probability densities. The technique is based on characterising hypotheses in terms of functionals which can be estimated root-n consistently, and constructing test statistics in terms of estimators of the functionals. Since the tests are semiparametric it is appropriate to assess them on the basis of their ability to detect departures of size n-1/2 from the null hypothesis. We show that they do indeed have this property. Unlike tests constructed in a nonparametric setting their power does not depend critically on choice of a bandwidth, and in particular, smoothing parameter selection is not an issue that has to be addressed by users of the tests. Bootstrap methods are suggested for calibrating the tests. In a regression setting, applications include tests of specification (such as partial linear and index models) against nonparametric or semiparametric alternatives, and tests of monotonicity, concavity, separability, equality of regression functions and base-independence of equivalence scales. In a density setting, they include tests of radial symmetry and stochastic dominance.