Testing Generalized Linear and Semiparametric Models Against Smooth Alternatives

We propose goodness of fit tests for testing generalized linear
models and semiparametric regression models against smooth
alternatives. The focus is on models having both, continuous and
factorial covariates. As smooth extension of a parametric or
semiparametric model we use generalized varying coefficient models
as proposed by Hastie&Tibshirani (JRSS B, 1993). A likelihood
ratio statistic is used for testing, and asymptotic normality of
the test statistic is proven. Due to a slow asymptotic convergence
rate a bootstrap approach is pursued. Asymptotic expansions allow
to write the estimates as linear smoothers which in turn guarantees
simple and fast bootstrapping. The test is shown to have sqrt(n)
power, but in contrast to parametric tests it is powerful against
smooth alternatives in general.