  Recently, Tibshirani et al. (2016) proposed a method for making inferences
about parameters defined by model selection, in a typical regression setting
with normally distributed errors. Here, we study the large sample properties of
this method, without assuming normality. We prove that the test statistic of
Tibshirani et al. (2016) is asymptotically valid, as the number of samples n
grows and the dimension d of the regression problem stays fixed. Our asymptotic
result holds uniformly over a wide class of nonnormal error distributions. We
also propose an efficient bootstrap version of this test that is provably
(asymptotically) conservative, and in practice, often delivers shorter
intervals than those from the original normality-based approach. Finally, we
prove that the test statistic of Tibshirani et al. (2016) does not enjoy
uniform validity in a high-dimensional setting, when the dimension d is allowed
grow.
