Statistics Papers

Document Type

Journal Article

Date of this Version

2016

Publication Source

The Annals of Statistics

Volume

44

Issue

3

Start Page

1038

Last Page

1068

DOI

10.1214/15-AOS1397

Abstract

We consider high-dimensional sparse regression problems in which we observe y = + z, where X is an n × p design matrix and z is an n-dimensional vector of independent Gaussian errors, each with variance σ2. Our focus is on the recently introduced SLOPE estimator [Ann. Appl. Stat. 9 (2015) 1103–1140], which regularizes the least-squares estimates with the rank-dependent penalty ∑1≤i≤pλi|βˆ|(i), where |βˆ|(i) is the ith largest magnitude of the fitted coefficients. Under Gaussian designs, where the entries of X are i.i.d. N(0,1/n), we show that SLOPE, with weights λi just about equal to σ⋅Φ−1(1−iq/(2p)) (Φ−1(α) is the αth quantile of a standard normal and q is a fixed number in (0,1)) achieves a squared error of estimation obeying sup‖β‖0≤kℙ(‖βˆSLOPEβ2>(1+ε)2σ2k log(p/k)) ⟶ 0 as the dimension p increases to ∞, and where ε>0 is an arbitrary small constant. This holds under a weak assumption on the sparsity level k and is sharp in the sense that this is the best possible error any estimator can achieve. A remarkable feature is that SLOPE does not require any knowledge of the degree of sparsity, and yet automatically adapts to yield optimal total squared errors over a wide range of sparsity classes. We are not aware of any other estimator with this property.

Copyright/Permission Statement

The original and published work is available at: https://projecteuclid.org/euclid.aos/1460381686#abstract

Keywords

SLOPE, Lasso, sparse regression, adaptivity, false discovery rate (FDR), Benjamini-Hochberg procedure, FDR thresholding

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Date Posted: 27 November 2017

This document has been peer reviewed.