## Electronic Journal of Statistics

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

Alexis Derumigny

#### Abstract

Extending the results of Bellec, Lecué and Tsybakov [1] to the setting of sparse high-dimensional 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 $(s/n)\log\left (p/s\right )$, 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 $l_{q}$-norm, with $q\in[1,2]$ for the Square-Root Lasso, and in the $l_{2}$ and sorted $l_{1}$ 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.

#### Article information

Source
Electron. J. Statist. Volume 12, Number 1 (2018), 741-766.

Dates
First available in Project Euclid: 27 February 2018

https://projecteuclid.org/euclid.ejs/1519722051

Digital Object Identifier
doi:10.1214/18-EJS1410

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62C20: Minimax procedures 62G05: Estimation

#### Citation

Derumigny, Alexis. Improved bounds for Square-Root Lasso and Square-Root Slope. Electron. J. Statist. 12 (2018), no. 1, 741--766. doi:10.1214/18-EJS1410. https://projecteuclid.org/euclid.ejs/1519722051

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