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

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  • Alexis Derumigny

    (CREST; ENSAE)

Abstract

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
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    References listed on IDEAS

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    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.
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