  Accurate statistical inference in logistic regression models remains a
critical challenge when the ratio between the number of parameters and sample
size is not negligible. This is because approximations based on either
classical asymptotic theory or bootstrap calculations are grossly off the mark.
This paper introduces a resized bootstrap method to infer model parameters in
arbitrary dimensions. As in the parametric bootstrap, we resample observations
from a distribution, which depends on an estimated regression coefficient
sequence. The novelty is that this estimate is actually far from the maximum
likelihood estimate (MLE). This estimate is informed by recent theory studying
properties of the MLE in high dimensions, and is obtained by appropriately
shrinking the MLE towards the origin. We demonstrate that the resized bootstrap
method yields valid confidence intervals in both simulated and real data
examples. Our methods extend to other high-dimensional generalized linear
models.
