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To estimate a sparse linear model from data with Gaussian noise, consiliencefrom lasso and compressed sensing literatures is that thresholding estimatorslike lasso and the Dantzig selector have the ability in some situations toidentify with high probability part of the significant covariatesasymptotically, and are numerically tractable thanks to convexity.  Yet, the selection of a threshold parameter $\lambda$ remains crucial inpractice. To that aim we propose Quantile Universal Thresholding, a selectionof $\lambda$ at the detection edge. We show with extensive simulations and realdata that an excellent compromise between high true positive rate and low falsediscovery rate is achieved, leading also to good predictive risk.

Quantile universal threshold: model selection at the detection edge for high-dimensional linear regression