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

Abstract

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, 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 $\sqrt{n}$(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.