Bayesian-motivated tests of function fit and their asymptotic frequentist properties

We propose and analyze nonparametric tests of the null hypothesis that a function belongs to a specified parametric family. The tests are based on BIC approximations, \pi_{BIC}, to the posterior probability of the null model, and may be carried out in either Bayesian or frequentist fashion. We obtain results on the asymptotic distribution of \pi_{BIC} under both the null hypothesis and local alternatives. One version of \pi_{BIC}, call it \pi_{BIC}^*, uses a class of models that are orthogonal to each other and growing in number without bound as sample size, n, tends to infinity. We show that \sqrtn(1-\pi_{BIC}^*) converges in distribution to a stable law under the null hypothesis. We also show that \pi_{BIC}^* can detect local alternatives converging to the null at the rate \sqrt\log n/n. A particularly interesting finding is that the power of the \pi_{BIC}^*-based test is asymptotically equal to that of a test based on the maximum of alternative log-likelihoods. Simulation results and an example involving variable star data illustrate desirable features of the proposed tests.