In this paper, we consider the estimation of regression coefficients and signal-to-noise (SNR) ratio in high-dimensional Generalized Linear Models (GLMs), and explore their implications in inferring popular estimands such as average treatment effects in high-dimensional observational studies. Under the ``proportional asymptotic'' regime and Gaussian covariates with known (population) covariance $\Sigma$, we derive Consistent and Asymptotically Normal (CAN) estimators of our targets of inference through a Method-of-Moments type of estimators that bypasses estimation of high dimensional nuisance functions and hyperparameter tuning altogether. Additionally, under non-Gaussian covariates, we demonstrate universality of our results under certain additional assumptions on the regression coefficients and $\Sigma$. We also demonstrate that knowing $\Sigma$ is not essential to our proposed methodology when the sample covariance matrix estimator is invertible. Finally, we complement our theoretical results with numerical experiments and comparisons with existing literature.

翻译：本文研究高维广义线性模型(GLM)中回归系数与信噪比(SNR)的估计问题，并探讨其在高维观测研究中推断常用估计量(如平均处理效应)的应用价值。在"比例渐近"框架下，针对协方差矩阵$\Sigma$已知的高斯协变量情形，我们通过矩估计类方法推导出具有一致性与渐近正态性(CAN)的估计量，该方法完全规避了高维冗余函数估计与超参数调优。此外，对于非高斯协变量情形，我们在回归系数与$\Sigma$满足特定附加假设的条件下证明了结果的普适性。我们同时证明，当样本协方差矩阵估计可逆时，$\Sigma$的先验知识对本方法并非必需。最后，我们通过数值实验以及与现有文献的对比验证了理论结果。