  In the research on checking whether the underlying model is of parametric
single-index structure with outliers in observations, the purpose of this paper
is two-fold. First, a test that is robust against outliers is suggested. The
Hampel's second-order influence function of the test statistic is proved to be
bounded. Second, the test fully uses the dimension reduction structure of the
hypothetical model and automatically adapts to alternative models when the null
hypothesis is false. Thus, the test can greatly overcome the dimensionality
problem and is still omnibus against general alternative models. The
performance of the test is demonstrated by both Monte Carlo simulation studies
and an application to a real dataset.
