Bayesian-motivated tests of function fit and their asymptotic frequentist properties
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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, 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 BIC under both the null hypothesis and local alternatives. One version of BIC, call it 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 n(1 - BIC*) converges in distribution to a stable law under the null hypothesis. We also show that BIC* can detect local alternatives converging to the null at the rate log n / n. A particularly interesting finding is that the power of the 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. Institute of Mathematical Statistics, 2004.