Pivotal estimation in high-dimensional regression via linear programming

Eric Gautier (1, 2) , Alexandre Tsybakov (1, 2)

1 CREST - Centre de Recherche en Économie et Statistique
2 ENSAE Paris - École Nationale de la Statistique et de l'Administration Économique

We propose a new method of estimation in high-dimensional linear regression model. It allows for very weak distributional assumptions including heteroscedasticity, and does not require the knowledge of the variance of random errors. The method is based on linear programming only, so that its numerical implementation is faster than for previously known techniques using conic programs, and it allows one to deal with higher dimensional models. We provide upper bounds for estimation and prediction errors of the proposed estimator showing that it achieves the same rate as in the more restrictive situation of fixed design and i.i.d. Gaussian errors with known variance. Following Gautier and Tsybakov (2011), we obtain the results under weaker sensitivity assumptions than the restricted eigenvalue or assimilated conditions.

Mots clés: Heteroscedasticity; High-dimensional models; Linear models; Model selection; Non-Gaussian errors; Pivotal estimation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]; Econométrie de la finance [q-fin.ST]

Eric Gautier, Alexandre Tsybakov. Pivotal estimation in high-dimensional regression via linear programming. 2013. ⟨hal-00805556v2⟩