## The Annals of Statistics

### CoCoLasso for high-dimensional error-in-variables regression

#### Abstract

Much theoretical and applied work has been devoted to high-dimensional regression with clean data. However, we often face corrupted data in many applications where missing data and measurement errors cannot be ignored. Loh and Wainwright [Ann. Statist. 40 (2012) 1637–1664] proposed a nonconvex modification of the Lasso for doing high-dimensional regression with noisy and missing data. It is generally agreed that the virtues of convexity contribute fundamentally the success and popularity of the Lasso. In light of this, we propose a new method named CoCoLasso that is convex and can handle a general class of corrupted datasets. We establish the estimation error bounds of CoCoLasso and its asymptotic sign-consistent selection property. We further elucidate how the standard cross validation techniques can be misleading in presence of measurement error and develop a novel calibrated cross-validation technique by using the basic idea in CoCoLasso. The calibrated cross-validation has its own importance. We demonstrate the superior performance of our method over the nonconvex approach by simulation studies.

#### Article information

Source
Ann. Statist. Volume 45, Number 6 (2017), 2400-2426.

Dates
Revised: August 2016
First available in Project Euclid: 15 December 2017

https://projecteuclid.org/euclid.aos/1513328577

Digital Object Identifier
doi:10.1214/16-AOS1527

Subjects
Primary: 62J07: Ridge regression; shrinkage estimators
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Datta, Abhirup; Zou, Hui. CoCoLasso for high-dimensional error-in-variables regression. Ann. Statist. 45 (2017), no. 6, 2400--2426. doi:10.1214/16-AOS1527. https://projecteuclid.org/euclid.aos/1513328577

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