the stimulating paper, which provides insights into statistical inference based on the lasso solution path. The authors proposed novel covariance statistics for testing the significance of predictor variables as they enter the active set, which formalizes the data-adaptive test based on the lasso path. The observation that “shrinkage ” balances “adaptivity ” to yield to an asymptotic Exp(1) null distribution is inspiring, and the mathematical analysis is delicate and intriguing. Adopting the notation from the paper under discussion, the main results are that the covariance statistics (Theorem 1) (Tk0+1,Tk0+2,...,Tk0+d) d → ( Exp(1), Exp(1/2),...,Exp(1/d)) (1) for orthogonal designs, and under the global null model (Theorem 2), T1 Exp(1), and under the general model (Theorem 3), P(Tk0+1 ≥ t) ≤ exp(−t) + o(1). These remarkable results are derived under a number of critical assumptions such as the normality, the sure screening [borrowing the terminology of Fan and Lv (2008)] or model selection consistency of the lasso path. As pointed out in Fa
