SLOPE-ADAPTIVE VARIABLE SELECTION VIA CONVEX OPTIMIZATION.

We introduce a new estimator for the vector of coefficients in the linear model = + , where has dimensions with possibly larger than . SLOPE, short for Sorted L-One Penalized Estimation, is the solution to [Formula: see text]where λ ≥ λ ≥ … ≥ λ ≥ 0 and [Formula: see text] are the decreasing absolute values of the entries of . This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical ℓ procedures such as the Lasso. Here, the regularizer is a sorted ℓ norm, which penalizes the regression coefficients according to their rank: the higher the rank-that is, stronger the signal-the larger the penalty. This is similar to the Benjamini and Hochberg [ (1995) 289-300] procedure (BH) which compares more significant -values with more stringent thresholds. One notable choice of the sequence {λ } is given by the BH critical values [Formula: see text], where ∈ (0, 1) and () is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λ provably controls FDR at level . Moreover, it also appears to have appreciable inferential properties under more general designs while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.