Comments on: High-dimensional simultaneous inference with the bootstrap

We congratulate the authors on their stimulating contribution to the burgeoning high-dimensional inference literature. The bootstrap offers such an attractive methodology in these settings, but it is well-known that its naive application in the context of shrinkage/superefficiency is fraught with danger (e.g. Samworth in Biometrika 90:985–990, 2003; Chatterjee and Lahiri in J Am Stat Assoc 106:608–625, 2011). The authors show how these perils can be elegantly sidestepped by working with de-biased, or de-sparsified, versions of estimators. In this discussion, we consider alternative approaches to individual and simultaneous inference in high-dimensional linear models, and retain the notation of the paper.