## Electronic Journal of Statistics

### Performance bounds for parameter estimates of high-dimensional linear models with correlated errors

#### Abstract

This paper develops a systematic theory for high-dimensional linear models with dependent errors and/or dependent covariates. To study properties of estimates of the regression parameters, we adopt the framework of functional dependence measures ([43]). For the covariates two schemes are addressed: the random design and the deterministic design. For the former we apply the constrained $\ell_{1}$ minimization approach, while for the latter the Lasso estimation procedure is used. We provide a detailed characterization on how the error rates of the estimates depend on the moment conditions that control the tail behaviors, the dependencies of the underlying processes that generate the errors and the covariates, the dimension and the sample size. Our theory substantially extends earlier ones by allowing dependent and/or heavy-tailed errors and the covariates. As our main tools, we derive exponential tail probability inequalities for dependent sub-Gaussian errors and Nagaev-type inequalities for dependent non-sub-Gaussian errors that arise from linear or non-linear processes.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 352-379.

Dates
First available in Project Euclid: 17 February 2016

https://projecteuclid.org/euclid.ejs/1455715966

Digital Object Identifier
doi:10.1214/16-EJS1108

Mathematical Reviews number (MathSciNet)
MR3466186

Zentralblatt MATH identifier
1333.62172

#### Citation

Wu, Wei-Biao; Wu, Ying Nian. Performance bounds for parameter estimates of high-dimensional linear models with correlated errors. Electron. J. Statist. 10 (2016), no. 1, 352--379. doi:10.1214/16-EJS1108. https://projecteuclid.org/euclid.ejs/1455715966

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