Confidence Sets Based on Thresholding Estimators in High-Dimensional Gaussian Regression Models
研究了高维线性回归中基于硬阈值、软阈值和自适应软阈值估计量的置信区间，分析了已知和未知误差方差下的覆盖性质，发现阈值估计量区间比最小二乘区间更大，且在一致变量选择时渐近大一个数量级。
We study confidence intervals based on hard-thresholding, soft-thresholding, and adaptive soft-thresholding in a linear regression model where the number of regressors k may depend on and diverge with sample size n. In addition to the case of known error variance, we define and study versions of the estimators when the error variance is unknown. In the known-variance case, we provide an exact analysis of the coverage properties of such intervals in finite samples. We show that these intervals are always larger than the standard interval based on the least-squares estimator. Asymptotically, the intervals based on the thresholding estimators are larger even by an order of magnitude when the estimators are tuned to perform consistent variable selection. For the unknown-variance case, we provide nontrivial lower bounds and a small numerical study for the coverage probabilities in finite samples. We also conduct an asymptotic analysis where the results from the known-variance case can be shown to carry over asymptotically if the number of degrees of freedom n − k tends to infinity fast enough in relation to the thresholding parameter.