IDEAS home Printed from
   My bibliography  Save this paper

Improved bounds for Square-Root Lasso and Square-Root Slope


  • Alexis Derumigny



Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse highdimensional linear regression with unknown variance, we show that two estimators, the Square-Root Lasso and the Square-Root Slope can achieve the optimal minimax prediction rate, which is (s/n) log (p/s), up to some constant, under some mild conditions on the design matrix. Here, n is the sample size, p is the dimension and s is the sparsity parameter. We also prove optimality for the estimation error in the lq-norm, with q in [1, 2] for the Square-Root Lasso, and in the l2 and sorted l1 norms for the Square-Root Slope. Both estimators are adaptive to the unknown variance of the noise. The Square-Root Slope is also adaptive to the sparsity s of the true parameter. Next, we prove that any estimator depending on s which attains the minimax rate admits an adaptive to s version still attaining the same rate. We apply this result to the Square-root Lasso. Moreover, for both estimators, we obtain valid rates for a wide range of confidence levels, and improved concentration properties as in [1] where the case of known variance is treated. Our results are non-asymptotic. ;Classification-JEL: Primary 62G08; secondary 62C20, 62G05.

Suggested Citation

  • Alexis Derumigny, 2017. "Improved bounds for Square-Root Lasso and Square-Root Slope," Working Papers 2017-53, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-53

    Download full text from publisher

    File URL:
    File Function: CREST working paper version
    Download Restriction: no

    References listed on IDEAS

    1. 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.
    2. Tingni Sun & Cun-Hui Zhang, 2012. "Scaled sparse linear regression," Biometrika, Biometrika Trust, vol. 99(4), pages 879-898.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Umberto Amato & Anestis Antoniadis & Italia De Feis & Irene Gijbels, 2021. "Penalised robust estimators for sparse and high-dimensional linear models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 1-48, March.
    2. Anindya Bhadra & Jyotishka Datta & Nicholas G. Polson & Brandon T. Willard, 2020. "Global-Local Mixtures: A Unifying Framework," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 82(2), pages 426-447, August.
    3. Beyhum, Jad, 2019. "Inference robust to outliers with L1‐norm penalization," TSE Working Papers 19-1032, Toulouse School of Economics (TSE).
    4. Zemin Zheng & Jinchi Lv & Wei Lin, 2021. "Nonsparse Learning with Latent Variables," Operations Research, INFORMS, vol. 69(1), pages 346-359, January.
    5. Victor Chernozhukov & Christian Hansen & Yuan Liao, 2015. "A lava attack on the recovery of sums of dense and sparse signals," CeMMAP working papers CWP56/15, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    6. Guo, Zijian & Kang, Hyunseung & Cai, T. Tony & Small, Dylan S., 2018. "Testing endogeneity with high dimensional covariates," Journal of Econometrics, Elsevier, vol. 207(1), pages 175-187.
    7. Jad Beyhum, 2020. "Inference robust to outliers with L1‐norm penalization," Post-Print hal-03235868, HAL.
    8. Sermpinis, Georgios & Tsoukas, Serafeim & Zhang, Ping, 2018. "Modelling market implied ratings using LASSO variable selection techniques," Journal of Empirical Finance, Elsevier, vol. 48(C), pages 19-35.
    9. Xie, Jichun & Kang, Jian, 2017. "High-dimensional tests for functional networks of brain anatomic regions," Journal of Multivariate Analysis, Elsevier, vol. 156(C), pages 70-88.
    10. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    11. Pun, Chi Seng & Hadimaja, Matthew Zakharia, 2021. "A self-calibrated direct approach to precision matrix estimation and linear discriminant analysis in high dimensions," Computational Statistics & Data Analysis, Elsevier, vol. 155(C).
    12. Jacob Bien & Irina Gaynanova & Johannes Lederer & Christian L. Müller, 2019. "Prediction error bounds for linear regression with the TREX," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 451-474, June.
    13. Jana Janková & Rajen D. Shah & Peter Bühlmann & Richard J. Samworth, 2020. "Goodness‐of‐fit testing in high dimensional generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 82(3), pages 773-795, July.
    14. Alexandre Belloni & Victor Chernozhukov & Denis Chetverikov & Christian Hansen & Kengo Kato, 2018. "High-dimensional econometrics and regularized GMM," CeMMAP working papers CWP35/18, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    15. Alexandre Belloni & Victor Chernozhukov & Kengo Kato, 2019. "Valid Post-Selection Inference in High-Dimensional Approximately Sparse Quantile Regression Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 749-758, April.
    16. Susan Athey & Guido W. Imbens & Stefan Wager, 2018. "Approximate residual balancing: debiased inference of average treatment effects in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(4), pages 597-623, September.
    17. Victor Chernozhukov & Ivan Fernandez-Val & Christian Hansen, 2013. "Program evaluation with high-dimensional data," CeMMAP working papers CWP57/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    18. Peter Bühlmann & Jacopo Mandozzi, 2014. "High-dimensional variable screening and bias in subsequent inference, with an empirical comparison," Computational Statistics, Springer, vol. 29(3), pages 407-430, June.
    19. Domenico Giannone & Michele Lenza & Giorgio E. Primiceri, 2021. "Economic Predictions With Big Data: The Illusion of Sparsity," Econometrica, Econometric Society, vol. 89(5), pages 2409-2437, September.
    20. Wang, Yihe & Zhao, Sihai Dave, 2021. "A nonparametric empirical Bayes approach to large-scale multivariate regression," Computational Statistics & Data Analysis, Elsevier, vol. 156(C).


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:crs:wpaper:2017-53. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Secretariat General (email available below). General contact details of provider: .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.