比例渐近下广义线性模型与双重稳健泛函的矩估计方法推断
Method-of-Moments Inference for GLMs and Doubly Robust Functionals under Proportional Asymptotics
AI总结：
本文在高维广义线性模型的比例渐近框架下提出矩估计方法，无需估计高维冗余函数或调参即可得到回归系数和信噪比的一致渐近正态估计，并推广至非高斯协变量及未知协方差情形，为平均处理效应等估计量推断提供理论支持。
摘要：
本文考虑高维广义线性模型（GLMs）中回归系数和信噪比（SNR）的估计问题，并探讨其在高维观测研究中推断平均处理效应等常用估计量时的意义。在“比例渐近”机制下，假设协变量为高斯分布且总体协方差Σ已知，我们通过一类矩估计方法推导出推断目标的一致渐近正态（CAN）估计量，该方法完全绕过了高维冗余函数的估计和超参数调优。此外，在非高斯协变量情形下，我们在回归系数和Σ满足一定额外假设时证明了结果的普适性。我们还证明，当样本协方差矩阵估计量可逆时，已知Σ并非所提方法的必要条件。最后，我们通过数值实验以及与现有文献的比较来补充理论结果。
英文摘要：
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 $Σ$, 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 $Σ$. We also demonstrate that knowing $Σ$ 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.
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