  Partial penalized tests provide flexible approaches to testing linear
hypotheses in high dimensional generalized linear models. However, because the
estimators used in these tests are local minimizers of potentially non-convex
folded-concave penalized objectives, the solutions one computes in practice may
not coincide with the unknown local minima for which we have nice theoretical
guarantees. To close this gap between theory and computation, we introduce
local linear approximation (LLA) algorithms to compute the full and reduced
model estimators for these tests and develop theory specifically for the LLA
solutions. We prove that our LLA algorithms converge to oracle estimators for
the full and reduced models in two steps with overwhelming probability. We then
leverage this strong oracle result and the asymptotic properties of the oracle
estimators to show that the partial penalized test statistics evaluated at the
two-step LLA solutions are approximately chi-square in large samples, giving us
guarantees for the tests using specific computed solutions and thereby closing
the theoretical gap. We conduct simulation studies to assess the finite-sample
performance of our testing procedures, finding that partial penalized tests
using the LLA solutions agree with tests using the oracle estimators, and
demonstrate our testing procedures in a real data application.
