2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130881We 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, π_{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 π_{BIC} under both the null hypothesis and local alternatives. One version of π_{BIC}, call it π_{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-π_{BIC}^*) converges in distribution to a stable law under the null hypothesis. We also show that π_{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 π_{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.Published at http://dx.doi.org/10.1214/009053604000000805 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)Statistics Theory62G10, 62C10 (Primary) 62G20. (Secondary)text