We consider the problem of inference for projection parameters in linear regression with increasing dimensions. This problem has been studied under a variety of assumptions in the literature. The classical asymptotic normality result for the least squares estimator of the projection parameter only holds when the dimension $d$ of the covariates is of smaller order than $n^{1/2}$, where $n$ is the sample size. Traditional sandwich estimator-based Wald intervals are asymptotically valid in this regime. In this work, we propose a bias correction for the least squares estimator and prove the asymptotic normality of the resulting debiased estimator as long as $d = o(n^{2/3})$, with an explicit bound on the rate of convergence to normality. We leverage recent methods of statistical inference that do not require an estimator of the variance to perform asymptotically valid statistical inference. We provide a discussion of how our techniques can be generalized to increase the allowable range of $d$ even further.

翻译：我们考虑在维度递增情形下线性回归中投影参数的推断问题。该问题已在文献中基于多种假设得到研究。经典的最小二乘估计量关于投影参数的渐近正态性结果仅在协变量维度 $d$ 远小于样本量 $n$ 的 $n^{1/2}$ 阶时成立。传统基于三明治估计量的Wald区间在此范围内具有渐近有效性。本文提出最小二乘估计量的偏差校正方法，并证明当 $d = o(n^{2/3})$ 时，所得去偏估计量的渐近正态性，同时给出向正态分布收敛速度的显式界。我们利用近期无需方差估计量即可实现渐近有效统计推断的方法，并讨论如何推广这些技术以进一步扩展 $d$ 的允许范围。