## Improved bounds for square-root Lasso and square-root slope.(English)Zbl 1473.62132

Summary: Extending the results of P. C. Bellec et al. [“Slope meets Lasso: improved oracle bounds and optimality”, Preprint, arXiv:1605.08651] 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 [loc. cit.] where the case of known variance is treated. Our results are non-asymptotic.

### MSC:

 62G08 Nonparametric regression and quantile regression 62C20 Minimax procedures in statistical decision theory 62G05 Nonparametric estimation 62J05 Linear regression; mixed models 62J07 Ridge regression; shrinkage estimators (Lasso)
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### References:

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