  To sufficiently exploit the model structure under the null hypothesis such
that the conditions on the whole model can be mild, this paper investigates
score function-based tests to check the significance of an
ultrahigh-dimensional sub-vector of the model coefficients when the nuisance
parameter vector is also ultrahigh-dimensional in linear models. We first
reanalyze and extend a recently proposed score function-based test to derive,
under weaker conditions, its limiting distributions under the null and local
alternative hypotheses. As it may fail to work when the correlation between
testing covariates and nuisance covariates is high, we propose an
orthogonalized score function-based test with two merits: debiasing to make the
non-degenerate error term degenerate and reducing the asymptotic variance to
enhance the power performance. Simulations evaluate the finite-sample
performances of the proposed tests, and a real data analysis illustrates its
application.
