Electronic Journal of Statistics

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

Wei-Biao Wu and Ying Nian Wu

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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.

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

Received: March 2015
First available in Project Euclid: 17 February 2016

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Consistency dependence-adjusted norm exponential inequality functional and predictive dependence measures high-dimensional time series impulse response function Nagaev inequality predictive persistence support recovery


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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