Annals of statistics vol:32 issue:6 pages:2580-2615
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 rootn(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 rootlogn/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.