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Testing a single regression coefficient in high dimensional linear models

机译：在高维线性模型中测试单个回归系数

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摘要

In linear regression models with high dimensional data, the classical z-test (or t-test) for testing the significance of each single regression coefficient is no longer applicable. This is mainly because the number of covariates exceeds the sample size. In this paper, we propose a simple and novel alternative by introducing the Correlated Predictors Screening (CPS) method to control for predictors that are highly correlated with the target covariate. Accordingly, the classical ordinary least squares approach can be employed to estimate the regression coefficient associated with the target covariate. In addition, we demonstrate that the resulting estimator is consistent and asymptotically normal even if the random errors are heteroscedastic. This enables us to apply the z-test to assess the significance of each covariate. Based on the p-value obtained from testing the significance of each covariate, we further conduct multiple hypothesis testing by controlling the false discovery rate at the nominal level. Then, we show that the multiple hypothesis testing achieves consistent model selection. Simulation studies and empirical examples are presented to illustrate the finite sample performance and the usefulness of the proposed method, respectively.
机译：在具有高维数据的线性回归模型中，用于测试每个回归系数的显着性的经典z检验（或t检验）不再适用。这主要是因为协变量的数量超过了样本量。在本文中，我们通过引入相关预测变量筛选（CPS）方法来控制与目标协变量高度相关的预测变量，从而提出了一种简单新颖的替代方案。因此，经典的普通最小二乘法可以用来估计与目标协变量相关的回归系数。此外，我们证明，即使随机误差是异方差的，所得的估计量也是一致且渐近正态的。这使我们能够应用z检验来评估每个协变量的显着性。基于从检验每个协变量的显着性获得的p值，我们通过将名义错误率控制在名义水平上，进一步进行了多个假设检验。然后，我们证明了多重假设检验可以实现一致的模型选择。仿真研究和实验实例分别说明了有限样本性能和所提方法的实用性。

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