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We study the estimation property of the Elastic Net estimator in high-dimensional linear regression models where the number of parameters  p  is comparable to or larger than the sample size  n . In such a situation, one often assumes sparsity of the true regression coefficient vector, i.e., assuming thatbelongs to an-ball with radius,, for some. In this paper, we provide-estimation error bounds for the Elastic Net and naive Elastic Net estimators under a unified framework for high-dimensional analysis of  M -estimators proposed by Negahban et al. [A unified framework for high-dimensional analysis of M-estimators with decomposable regularizers. Adv Neural Inf Process Syst. 2009;22:1348–1356]. We show that for both cases of exact sparsity and weak sparsity, under the same conditions on the design matrix, the Elastic Net estimator achieves a slightly better error bound than the Lasso estimator by suitably choosing the tuning parameters.

On estimation error bounds of the Elastic Net when p ≫ n