##
**On estimation of nonsmooth functionals of sparse normal means.**
*(English)*
Zbl 1442.62078

The authors establish non-asymptotic minimax optimal rates of estimation on the classes of sparse vectors and proposed estimators achieving these rates. They proposed estimators for the quantities \(N_\gamma (\theta) = \sum_{i=1}^d|\theta_i|^\gamma\), \(\gamma>0\) and \(\ell_\gamma (\theta)\), \(\gamma \geq 1\), where \(\ell_\gamma\) denotes the norm of order \(\gamma\).

The authors realized that, for the general sparsity classes \(B_0 (s)\), where \[ B_0(s) = \{\boldsymbol{\theta} \in \mathbb{R}^d: || \boldsymbol{\theta} ||_0 \leq s\} \] and \(|| \boldsymbol{\theta} ||_{0}\) denotes the number of nonzero components of \(\boldsymbol{\theta} \mbox{ and } s \in \left\lbrace 1, \dots, d\right\rbrace\), there exist two different regimes with an elbow at \(s \asymp \sqrt{d}\) and they called them the sparse zone and the dense zone. In the paper, the dense zone is defined in the \(s^{2} \geq 4d\) section while the sparse zone is defined in the complementary section, that is, \(s^{2} < 4d\).

The authors proposed two estimators for the dense zone, one for the case where \(\gamma\) is an even integer and the another for the other cases. The authors explained that the estimator for the case where \(\gamma\) is an even integer is justified by the fact that the even power functionals \(N_{\gamma}(\boldsymbol{\theta})\) admit unbiased estimators converging at rates much faster than the estimators for the others \(\gamma\) values, for which the functionals \(N_{\gamma}(\boldsymbol{\theta})\) are not smooth.

An interesting aspect of the method of obtaining the estimator for the case where \(\gamma\) is not an even integer is that the authors use a sample duplication technique. Another important observation is that the proposed estimator for the case where \(\gamma\) is an even integer is also valid for the context of the sparse zone, but the authors found even better estimators in the context of sparse zone that achieve the optimal rate.

The authors realized that, for the general sparsity classes \(B_0 (s)\), where \[ B_0(s) = \{\boldsymbol{\theta} \in \mathbb{R}^d: || \boldsymbol{\theta} ||_0 \leq s\} \] and \(|| \boldsymbol{\theta} ||_{0}\) denotes the number of nonzero components of \(\boldsymbol{\theta} \mbox{ and } s \in \left\lbrace 1, \dots, d\right\rbrace\), there exist two different regimes with an elbow at \(s \asymp \sqrt{d}\) and they called them the sparse zone and the dense zone. In the paper, the dense zone is defined in the \(s^{2} \geq 4d\) section while the sparse zone is defined in the complementary section, that is, \(s^{2} < 4d\).

The authors proposed two estimators for the dense zone, one for the case where \(\gamma\) is an even integer and the another for the other cases. The authors explained that the estimator for the case where \(\gamma\) is an even integer is justified by the fact that the even power functionals \(N_{\gamma}(\boldsymbol{\theta})\) admit unbiased estimators converging at rates much faster than the estimators for the others \(\gamma\) values, for which the functionals \(N_{\gamma}(\boldsymbol{\theta})\) are not smooth.

An interesting aspect of the method of obtaining the estimator for the case where \(\gamma\) is not an even integer is that the authors use a sample duplication technique. Another important observation is that the proposed estimator for the case where \(\gamma\) is an even integer is also valid for the context of the sparse zone, but the authors found even better estimators in the context of sparse zone that achieve the optimal rate.

Reviewer: Kévin Allan Sales Rodrigues (São Paulo)

### Keywords:

functional estimation; nonsmooth functional; norm estimation; polynomial approximation; sparsity### References:

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