## Bernoulli

- Bernoulli
- Volume 25, Number 3 (2019), 1901-1938.

### Sparse covariance matrix estimation in high-dimensional deconvolution

Denis Belomestny, Mathias Trabs, and Alexandre B. Tsybakov

#### Abstract

We study the estimation of the covariance matrix $\Sigma$ of a $p$-dimensional normal random vector based on $n$ independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of $\Sigma$. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in $n/\log p$. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.

#### Article information

**Source**

Bernoulli, Volume 25, Number 3 (2019), 1901-1938.

**Dates**

Received: October 2017

Revised: March 2018

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1560326432

**Digital Object Identifier**

doi:10.3150/18-BEJ1040A

**Mathematical Reviews number (MathSciNet)**

MR3961235

**Zentralblatt MATH identifier**

07066244

**Keywords**

Fourier methods minimax convergence rates severely ill-posed inverse problem thresholding

#### Citation

Belomestny, Denis; Trabs, Mathias; Tsybakov, Alexandre B. Sparse covariance matrix estimation in high-dimensional deconvolution. Bernoulli 25 (2019), no. 3, 1901--1938. doi:10.3150/18-BEJ1040A. https://projecteuclid.org/euclid.bj/1560326432

#### Supplemental materials

- Supplement: A bound for weighted $L^{2}$-distances of certain densities. We prove Proposition 16 which is needed to show the lower bound for the covariance estimation in the deconvolution model. More precisely, a bound for the $L^{2}$-distance of the certain densities with polynomial weights is proved.Digital Object Identifier: doi:10.3150/18-BEJ1040ASUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.