  Local smoothing testing that is based on multivariate nonparametric
regression estimation is one of the main model checking methodologies in the
literature. However, relevant tests suffer from the typical curse of
dimensionality resulting in slow convergence rates to their limits under the
null hypotheses and less deviation from the null under alternatives. This
problem leads tests to not well maintain the significance level and to be less
sensitive to alternatives. In this paper, a dimension-reduction model-adaptive
test is proposed for generalized linear models. The test behaves like a local
smoothing test as if the model were univariate, and can be consistent against
any global alternatives and can detect local alternatives distinct from the
null at a fast rate that existing local smoothing tests can achieve only when
the model is univariate. Simulations are carried out to examine the performance
of our methodology. A real data analysis is conducted for illustration. The
method can readily be extended to global smoothing methodology and other
testing problems.
