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Linear and conic programming estimators in high dimensional errors‐in‐variables models
Author(s) -
Belloni Alexandre,
Rosenbaum Mathieu,
Tsybakov Alexandre B.
Publication year - 2017
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12196
Subject(s) - minimax , estimator , linear programming , mathematics , mathematical optimization , minimax estimator , lasso (programming language) , linear model , conic section , linear regression , computer science , algorithm , minimum variance unbiased estimator , statistics , geometry , world wide web
Summary We consider the linear regression model with observation error in the design. In this setting, we allow the number of covariates to be much larger than the sample size. Several new estimation methods have been recently introduced for this model. Indeed, the standard lasso estimator or Dantzig selector turns out to become unreliable when only noisy regressors are available, which is quite common in practice. In this work, we propose and analyse a new estimator for the errors‐in‐variables model. Under suitable sparsity assumptions, we show that this estimator attains the minimax efficiency bound. Importantly, this estimator can be written as a second‐order cone programming minimization problem which can be solved numerically in polynomial time. Finally, we show that the procedure introduced by Rosenbaum and Tsybakov, which is almost optimal in a minimax sense, can be efficiently computed by a single linear programming problem despite non‐convexities.