  Nonparametric generalized likelihood ratio test is popularly used for model
checking for regressions. However, there are two issues that may be the
barriers for its powerfulness. First, the bias term in its liming null
distribution causes the test not to well control type I error and thus Monte
Carlo approximation for critical value determination is required. Second, it
severely suffers from the curse of dimensionality due to the use of
multivariate nonparametric function estimation. The purpose of this paper is
thus two-fold: a bias-correction is suggested to this test and a dimension
reduction-based model-adaptive enhancement is recommended to promote the power
performance. The proposed test still possesses the Wilks phenomenon, and the
test statistic can converge to its limit at a much faster rate and is much more
sensitive to alternative models than the original nonparametric generalized
likelihood ratio test as if the dimension of covariates were one. Simulation
studies are conducted to evaluate the finite sample performance and to compare
with other popularly used tests. A real data analysis is conducted for
illustration.

Methodology