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（GLMs）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见估计量（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，我们通过矩量法类估计量推导出目标参数的相合渐近正态（CAN）估计量，该方法完全规避了高维冗余函数的估计与超参数调优。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论的普适性。同时，我们论证了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，我们通过数值实验及与现有文献的对比研究对理论结果进行了补充验证。