Frequentist-Bayes Lack-of-Fit Tests Based on Laplace Approximations

The null hypothesis that all of a function’s Fourier coefficients are 0 is tested in frequentist fashion using as test statistic a Laplace approximation to the posterior probability of the null hypothesis. Testing whether or not a regression function has a prescribed linear form is one application of such a test. In contrast to BIC, the Laplace approximation depends on prior probabilities, and hence allows the investigator to tailor the test to particular kinds of alternative regression functions. On the other hand, using diffuse priors produces new omnibus lack-of-fit statistics. The new omnibus test statistics are weighted sums of exponentiated squared (and normalized) Fourier coefficients, where the weights depend on prior probabilities. Exponentiation of the Fourier components leads to tests that can be exceptionally powerful against high frequency alternatives. Evidence to this effect is provided by a comprehensive simulation study, in which one new test that had good power at high frequencies also performed comparably to some other well-known omnibus tests at low frequency alternatives.