  Residual marked empirical process-based tests are commonly used in regression
models. However, they suffer from data sparseness in high-dimensional space
when there are many covariates. This paper has three purposes. First, we
suggest a partial dimension reduction adaptive-to-model testing procedure that
can be omnibus against general global alternative models although it fully use
the dimension reduction structure under the null hypothesis. This feature is
because that the procedure can automatically adapt to the null and alternative
models, and thus greatly overcomes the dimensionality problem. Second, to
achieve the above goal, we propose a ridge-type eigenvalue ratio estimate to
automatically determine the number of linear combinations of the covariates
under the null and alternatives. Third, a Monte-Carlo approximation to the
sampling null distribution is suggested. Unlike existing bootstrap
approximation methods, this gives an approximation as close to the sampling
null distribution as possible by fully utilising the dimension reduction model
structure under the null. Simulation studies and real data analysis are then
conducted to illustrate the performance of the new test and compare it with
existing tests.
