论文标题
与隐藏变量的高维多元响应回归的推断
Inference in High-dimensional Multivariate Response Regression with Hidden Variables
论文作者
论文摘要
本文研究了在存在隐藏变量的情况下，在多元响应线性回归下的回归系数基质的推断。提出了一种构建系数矩阵条目置信区间的新过程。我们的方法首先通过估计和调整隐藏效果来构建系数矩阵的初始估计器，从而利用响应的多元性质。通过进一步部署一个低维投影程序，以减少上一步中正则化引入的偏差，提出了一个精制的估计量，并证明是渐近正常的。所得估计量的渐近方差以封闭形式的表达得出，并且可以始终如一地进行估计。此外，我们为存在隐藏效应的存在提出了一个测试程序，并提供了其理论上的理由。即使特征维度和响应数量超过样本量，我们的过程及其分析都是有效的。我们的结果通过广泛的模拟和实际数据分析进一步备份。
This paper studies the inference of the regression coefficient matrix under multivariate response linear regressions in the presence of hidden variables. A novel procedure for constructing confidence intervals of entries of the coefficient matrix is proposed. Our method first utilizes the multivariate nature of the responses by estimating and adjusting the hidden effect to construct an initial estimator of the coefficient matrix. By further deploying a low-dimensional projection procedure to reduce the bias introduced by the regularization in the previous step, a refined estimator is proposed and shown to be asymptotically normal. The asymptotic variance of the resulting estimator is derived with closed-form expression and can be consistently estimated. In addition, we propose a testing procedure for the existence of hidden effects and provide its theoretical justification. Both our procedures and their analyses are valid even when the feature dimension and the number of responses exceed the sample size. Our results are further backed up via extensive simulations and a real data analysis.
