Open Access
December 2004 Bayesian-motivated tests of function fit and their asymptotic frequentist properties
Marc Aerts, Gerda Claeskens, Jeffrey D. Hart
Ann. Statist. 32(6): 2580-2615 (December 2004). DOI: 10.1214/009053604000000805


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 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.


Download Citation

Marc Aerts. Gerda Claeskens. Jeffrey D. Hart. "Bayesian-motivated tests of function fit and their asymptotic frequentist properties." Ann. Statist. 32 (6) 2580 - 2615, December 2004.


Published: December 2004
First available in Project Euclid: 7 February 2005

zbMATH: 1068.62053
MathSciNet: MR2153996
Digital Object Identifier: 10.1214/009053604000000805

Primary: 62C10 , 62G10
Secondary: 62G20

Keywords: asymptotic distribution , BIC , local alternatives , nonparametric lack of-fit-test , orthogonal series , stable distribution

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 6 • December 2004
Back to Top