Asymptotic properties on high-dimensional multivariate regression M-estimation

Hao Ding, Shanshan Qin, Yuehua Wu and Yaohua Wu

Journal of Multivariate Analysis, 2021, vol. 183, issue C

Abstract: In this paper, we work on a general multivariate regression model under the regime that both p, the number of covariates, and n, the number of observations, are large with p∕n→κ(0<κ<∞). Unlike previous works that focus on a sparse regression vector β, we consider a more interesting situation in which β is composed of two groups: components in group I are large while components in group II are small but possibly not zeros. This study aims to explore the asymptotic behavior of the ridge-regularized high-dimensional multivariate M-estimator of β in group II. By applying the double leave-one-out method, we successfully derive a nonlinear system comprised of two deterministic equations, which characterizes the risk behavior of the M-estimator. The system solution also enables us to yield asymptotic normality for each component of the M-estimator. Moreover, we present rigorous proofs to these approximations that play a critical role in deriving the system. Finally, we perform experimental validations to demonstrate the performance of the proposed system.

Keywords: Double leave-one-out method; High-dimensional; M-estimation; Multivariate regression; Nonlinear system; Proximal mapping (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1016/j.jmva.2021.104730

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