We are grateful for the opportunity to discuss this new test, based on marginal screening, of a global null hypothesis in linear models. Marginal screening has become a very popular tool for reducing dimensionality in recent years, and a great deal of work has focused on its variable selection properties (e.g., Fan and Lv 2008; Fan, Samworth, and Wu 2009). Corresponding inference procedures are much less well developed, and one of the interesting contributions of this article is the observation that the limiting distribution (here and throughout, we use the same notation as in the article) of n1/2(θˆn−θ0) is discontinuous at θ0 = 0. Such nonregular limiting distributions are well known to cause difficulties for the bootstrap (e.g., Beran 1997; Samworth 2003). Although in some settings, these issues are an artefact of the pointwise asymptotics of consistency usually invoked to justify the bootstrap (Samworth 2005), there are other settings where some modification of standard bootstrap procedures is required. Two such examples include bootstrapping Lasso estimators (Chatterjee and Lahiri 2011) and certain classification problems (Laber and Murphy 2011), where thresholded versions of the obvious estimators are bootstrapped, in an analogous fashion to the approach in this article.
Leverhulme Trust (PLP-2014-353)