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We propose and analyze nonparametric tests of the null hypothesis that afunction belongs to a specified parametric family. The tests are based on BICapproximations, \pi_{BIC}, to the posterior probability of the null model, andmay be carried out in either Bayesian or frequentist fashion. We obtain resultson the asymptotic distribution of \pi_{BIC} under both the null hypothesis andlocal alternatives. One version of \pi_{BIC}, call it \pi_{BIC}^*, uses a classof models that are orthogonal to each other and growing in number without boundas 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 alsoshow that \pi_{BIC}^* can detect local alternatives converging to the null atthe rate \sqrt\log n/n. A particularly interesting finding is that the power ofthe \pi_{BIC}^*-based test is asymptotically equal to that of a test based onthe maximum of alternative log-likelihoods. Simulation results and an exampleinvolving variable star data illustrate desirable features of the proposedtests.

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