Regularization methods, including Lasso, group Lasso and SCAD, typically focus on selecting variables with strong effects while ignoring weak signals. This may result in biased prediction, especially when weak signals outnumber strong signals. This paper aims to incorporate weak signals in variable selection, estimation and prediction. We propose a two-stage procedure, consisting of variable selection and post-selection estimation. The variable selection stage involves a covariance-insured screening for detecting weak signals, while the post-selection estimation stage involves a shrinkage estimator for jointly estimating strong and weak signals selected from the first stage. We term the proposed method as the covariance-insured screening based post-selection shrinkage estimator. We establish asymptotic properties for the proposed method and show, via simulations, that incorporating weak signals can improve estimation and prediction performance. We apply the proposed method to predict the annual gross domestic product (GDP) rates based on various socioeconomic indicators for 82 countries.

中文翻译：

---

高维回归中的弱信号：检测、估计和预测

正则化方法，包括 Lasso、group Lasso 和 SCAD，通常侧重于选择具有强影响的变量而忽略弱信号。这可能会导致有偏差的预测，尤其是当弱信号多于强信号时。本文旨在将弱信号纳入变量选择、估计和预测中。我们提出了一个两阶段程序，包括变量选择和选择后估计。变量选择阶段涉及用于检测弱信号的协方差保证筛选，而后选择估计阶段涉及用于联合估计从第一阶段选择的强弱信号的收缩估计器。我们将所提出的方法称为基于协方差保险筛选的选择后收缩估计器。我们为所提出的方法建立渐近性质，并表明，通过模拟，结合弱信号可以提高估计和预测性能。我们应用所提出的方法根据 82 个国家的各种社会经济指标预测年度国内生产总值 (GDP) 率。