主 题：Estimation of Linear Functionals in High Dimensional Linear Models:

From Sparsity to Non-sparsity（高维线性模型中线性函数函的估计：从稀疏到非稀疏）

主讲人：北京师范大学赵俊龙教授

主持人：统计学院林华珍教授

时间：2023年5月26日（周五）下午16:00-17:00

举办地点：柳林校区弘远楼408会议室

主办单位：统计研究中心和统计学院 科研处

主讲人简介：

赵俊龙，北京师范大学统计学院教授。 从事统计和机器学习相关研究，包括：高维数据分析、统计机器学习、稳健统计等。 在统计学顶级期刊Journal of the Royal Statistical Society: Series B（JRSSB）、 The Annals of Statistics（AOS）、Journal of American Statistical Association(JASA)，Biometrika以及其他统计学重要期刊发了论文四十余篇。主持多项国家自然科学基金项目，参与国家自然科学基金重点项目。

内容简介：

High dimensional linear models are commonly used in practice. In many applications, one is interested in linear transformations $\bbeta^\top x$ of regression coefficients $\bbeta\ in \mR^p$, where $x$ is a specific point and is not required to be identically distributed as the training data. One common approach is the plug-in technique which first estimates $\bbeta$, then plugs the estimator in the linear transformation for prediction. Despite its popularity, estimation of $\bbeta$ can be difficult for high dimensional problems. Commonly used assumptions in the literature include that the signal of coefficients $\bbeta$ is sparse and predictors are weakly correlated. These assumptions, however, may not be easily verified, and can be violated in practice. When $\bbeta$ is non-sparse or predictors are strongly correlated, estimation of $\bbeta$ can be very difficult. In this paper, we propose a novel pointwise estimator for linear transformations of $\bbeta$. This new estimator greatly relaxes the common assumptions for high dimensional problems, and is adaptive to the degree of sparsity of $\bbeta$ and strength of correlations among the predictors. In particular，$\bbeta$ can be sparse or non-sparse and predictors can be strongly or weakly correlated. The proposed method is simple for implementation. Numerical and theoretical results demonstrate the competitive advantages of the proposed method for a wide range of problems.

高维线性模型在实践中常用。在许多应用中，人们对中\mR^p$空间中回归系数$\bbeta\ \top x$的线性变换$\bbeta\ \感兴趣，其中$x$是一个特定的点，不需要与训练数据同分布。一种常见的方法是plug-in技术，它首先估计$\bbeta$，然后将估计量插入线性变换中进行预测。尽管它很受欢迎，但对于高维问题来说，估计$\bbeta$可能很困难。文献中常用的假设包括系数 $\bbeta$ 的信号稀疏且预测变量相关性较弱。然而，这些假设可能不容易验证，并且在实践中可能会被违反。当$\bbeta$非稀疏或预测变量高度相关时，估计$\bbeta$可能非常困难。在本文中，主讲人提出了一种新的$\bbeta$线性变换的逐点估计量。这个新的估计量大大放宽了高维问题的常见假设，并且对$\bbeta$的稀疏程度和预测变量间的相关性强度具有适应性。特别是，  可以是稀疏的或非稀疏的，预测变量可以是强相关或弱相关。所提出的方法易于实现。数值和理论结果表明，所提方法在解决各种问题时具有竞争优势。