[[ Edit: deleted question about what 'k' is for the discretized 1{ rank <= k } response. It's mentioned in the article ]]
I'm genuinely curious how to do coef significance testing for L1-regularized models. I once saw someone ask this at a Tibshirani talk and he said "oh we have no idea, we've resorted to the bootstrap before".
% of replicates with (coef==0) is potentially much more clever, especially since that's the test we want to perform anyway. i'll run that over the data and see what changes.
From a frequentist perspective, counting zeroes don't make much sense because under the null of coef=0 there is still a chance you don't estimate coef=0, even after regularization.
I think the question is these don't look like NormalCDF(coef/se) p-values given the coef and se you report. They tend to be too small.