  After variable selection, standard inferential procedures for regression
parameters may not be uniformly valid; there is no finite-sample size at which
a standard test is guaranteed to approximately attain its nominal size. This
problem is exacerbated in high-dimensional settings, where variable selection
becomes unavoidable. This has prompted a flurry of activity in developing
uniformly valid hypothesis tests for a low-dimensional regression parameter
(e.g. the causal effect of an exposure A on an outcome Y) in high-dimensional
models. So far there has been limited focus on model misspecification, although
this is inevitable in high-dimensional settings. We propose tests of the null
that are uniformly valid under sparsity conditions weaker than those typically
invoked in the literature, assuming working models for the exposure and outcome
are both correctly specified. When one of the models is misspecified, by
amending the procedure for estimating the nuisance parameters, our tests
continue to be valid; hence they are doubly robust. Our proposals are
straightforward to implement using existing software for penalized maximum
likelihood estimation and do not require sample-splitting. We illustrate them
in simulations and an analysis of data obtained from the Ghent University
Intensive Care Unit.
