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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 [Formula: see text]where λ1 ≥ λ2 ≥ … ≥ λ p ≥ 0 and [Formula: see text] are the decreasing absolute values of the entries of b. This is(More)
We derive a second-order ordinary differential equation (ODE), which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's scheme. As a byproduct, we obtain a family(More)
We introduce a novel method for sparse regression and variable selection, which is inspired by modern ideas in multiple testing. Imagine we have observations from the linear model y = Xβ + z, then we suggest estimating the regression coefficients by means of a new estimator called SLOPE, which is the solution to minimize b 1 2 y − Xb 2 2 + λ 1 |b| (1) + λ 2(More)
We introduce a novel method for sparse regression and variable selection, which is inspired by modern ideas in multiple testing. Imagine we have observations from the linear model y = Xβ + z, then we suggest estimating the regression coefficients by means of a new estimator called the ordered lasso, which is the solution to minimize b 1 2 y − Xb 2 2 + λ 1(More)
The epithelium of the intestinal mucosa is a rapidly self-renewing tissue in the body, and defects in the renewal process occur commonly in various disorders. microRNAs (miRNAs) posttranscriptionally regulate gene expression and are implicated in many aspects of cellular physiology. Here we investigate the role of miRNA-29b (miR-29b) in the regulation of(More)
We consider high-dimensional sparse regression problems in which we observe y = Xβ + z, where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussian errors, each with variance σ 2. Our focus is on the recently introduced SLOPE estimator [15], which regularizes the least-squares estimates with the rank-dependent penalty 1≤i≤p λ(More)
We present a novel method for controlling the k-familywise error rate (k-FWER) in the linear regression setting using the knockoffs framework first introduced by Barber and Candès. Our procedure, which we also refer to as knockoffs, can be applied with any design matrix with at least as many observations as variables, and does not require knowing the noise(More)
In regression settings where explanatory variables have very low correlations and where there are relatively few effects each of large magnitude, it is commonly believed that the Lasso shall be able to find the important variables with few errors—if any. In contrast, this paper shows that this is not the case even when the design variables are(More)
In this note we give a proof showing that even though the number of false discoveries and the total number of discoveries are not continuous functions of the parameters, the formulas we obtain for the false discovery proportion (FDP) and the power, namely, (B.3) and (B.4) in the paper Statistical Estimation and Testing via the Sorted 1 Norm are(More)