We develop a new method to estimate the projection direction in the debiased Lasso estimator. The basic idea is to decompose the overall bias into two terms corresponding to strong and weak signals respectively. We propose to estimate the projection direction by balancing the squared biases associated with the strong and weak signals as well as the variance of the projection-based estimator. Standard quadratic programming solver can efficiently solve the resulting optimization problem. In theory, we show that the unknown set of strong signals can be consistently estimated and the projection-based estimator enjoys the asymptotic normality under suitable assumptions. A slight modification of our procedure leads to an estimator with a potentially smaller order of bias comparing to the original debiased Lasso. We further generalize our method to conduct inference for a sparse linear combination of the regression coefficients. Numerical studies demonstrate the advantage of the proposed approach concerning coverage accuracy over some existing alternatives.

中文翻译：

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高维线性模型的基于投影的推理

我们开发了一种新方法来估计去偏 Lasso 估计器中的投影方向。基本思想是将整体偏差分解为分别对应强信号和弱信号的两项。我们建议通过平衡与强信号和弱信号相关的平方偏差以及基于投影的估计器的方差来估计投影方向。标准的二次规划求解器可以有效地解决由此产生的优化问题。从理论上讲，我们表明可以一致地估计一组未知的强信号，并且基于投影的估计器在适当的假设下享有渐近正态性。对我们的程序稍作修改，就可以得到一个估计器，与原始去偏的 Lasso 相比，其偏差阶数可能更小。我们进一步推广我们的方法来对回归系数的稀疏线性组合进行推理。数值研究证明了所提出的方法在覆盖精度方面优于一些现有替代方法。