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 $p∕n\to \kappa (0<\kappa <\infty )$. Unlike previous works that focus on a sparse regression vector $\mathit{\beta }$, we consider a more interesting situation in which $\mathit{\beta }$ 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 $\mathit{\beta }$ 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.

在本文中，我们研究了在以下两种情况下的通用多元回归模型： $p$，协变量的数量，以及 $ñ$，观察的数量很大， $p∕ñ\to \kappa （0<\kappa <\infty ）$。与以前的工作着眼于稀疏回归向量不同$\mathit{\beta }$，我们认为一种更有趣的情况是 $\mathit{\beta }$由两部分组成：第一组中的分量很大，而第二组中的分量很小，但可能不是零。这项研究旨在探讨正则化的脊正则化高维多元M估计量的渐近行为$\mathit{\beta }$在第二组。通过应用双重留一法，我们成功地得出了一个由两个确定性方程组成的非线性系统，该系统表征了M估计量的风险行为。该系统解决方案还使我们能够为M估计量的每个分量产生渐近正态性。此外，我们为这些近似值提供了严格的证明，这些证明在推导系统中起着至关重要的作用。最后，我们进行实验验证以证明所提出系统的性能。