We introduce a new estimator for the vector of coefficients β in the
linear model y=Xβ+z, where X has dimensions n×p with p
possibly larger than n. SLOPE, short for Sorted L-One Penalized Estimation,
is the solution to b∈Rpmin21∥y−Xb∥ℓ22+λ1∣b∣(1)+λ2∣b∣(2)+⋯+λp∣b∣(p), where
λ1≥λ2≥⋯≥λp≥0 and ∣b∣(1)≥∣b∣(2)≥⋯≥∣b∣(p) are the
decreasing absolute values of the entries of b. This is a convex program and
we demonstrate a solution algorithm whose computational complexity is roughly
comparable to that of classical ℓ1 procedures such as the Lasso. Here,
the regularizer is a sorted ℓ1 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 [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] procedure (BH) which
compares more significant p-values with more stringent thresholds. One
notable choice of the sequence {λi} is given by the BH critical
values λBH(i)=z(1−i⋅q/2p), where q∈(0,1) and
z(α) 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 λBH provably controls FDR at level q.
Moreover, it also appears to have appreciable inferential properties under more
general designs X while having substantial power, as demonstrated in a series
of experiments running on both simulated and real data.Comment: Published at http://dx.doi.org/10.1214/15-AOAS842 in the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org