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, πBIC, 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 πBIC under both the null hypothesis and local alternatives. One version of πBIC, call it πBIC*, 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 $\sqrt{n}$ (1−πBIC*) converges in distribution to a stable law under the null hypothesis. We also show that πBIC* 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 πBIC*-based test is asymptotically equal to that of a test based on the maximum of alternative log-likelihoods.
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Simulation results and an example involving variable star data illustrate desirable features of the proposed tests.