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Pivotal estimation in high-dimensional regression via linear programming

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  • Eric Gautier

    ()
    (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique, ENSAE - École Nationale de la Statistique et de l'Administration Économique - ENSAE ParisTech)

  • Alexandre Tsybakov

    ()
    (CREST - Centre de Recherche en Économie et Statistique - INSEE - École Nationale de la Statistique et de l'Administration Économique, ENSAE - École Nationale de la Statistique et de l'Administration Économique - ENSAE ParisTech)

Abstract

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.

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File URL: http://hal.archives-ouvertes.fr/docs/00/81/32/06/PDF/GautierTsybakov_AdptDantzigp.pdf
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Bibliographic Info

Paper provided by HAL in its series Working Papers with number hal-00805556.

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Date of creation: 26 Mar 2013
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Handle: RePEc:hal:wpaper:hal-00805556

Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00805556
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Related research

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

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  1. A. Belloni & D. Chen & Victor Chernozhukov & Christian Hansen, 2010. "Sparse models and methods for optimal instruments with an application to eminent domain," CeMMAP working papers CWP31/10, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  2. Eric Gautier & Alexandre Tsybakov, 2011. "High-dimensional instrumental variables regression and confidence sets," Working Papers hal-00591732, HAL.
  3. A. Belloni & V. Chernozhukov & L. Wang, 2011. "Square-root lasso: pivotal recovery of sparse signals via conic programming," Biometrika, Biometrika Trust, vol. 98(4), pages 791-806.
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