We developed a statistical inference method applicable to a broad range of generalized linear models (GLMs) in high-dimensional settings, where the number of unknown coefficients scales proportionally with the sample size. Although a pioneering inference method has been developed for logistic regression, which is a specific instance of GLMs, we cannot apply this method directly to other GLMs because of unknown hyper-parameters. In this study, we addressed this limitation by developing a new inference method designed for a certain class of GLMs. Our method is based on the adjustment of asymptotic normality in high dimensions and is feasible in the sense that it is possible even with unknown hyper-parameters. Specifically, we introduce a novel convex loss-based estimator and its associated system, which are essential components of inference. Next, we devise a moment-based method for estimating the system parameters required by the method. Consequently, we construct confidence intervals for GLMs in a high-dimensional regime. We prove that our proposed method has desirable theoretical properties, such as strong consistency and exact coverage probability. Finally, we experimentally confirmed its validity.

翻译：我们开发了一种适用于高维环境下广泛广义线性模型（GLMs）的统计推断方法，其中未知系数数量与样本量成比例增长。尽管已有针对逻辑回归（GLMs的一个特例）的开创性推断方法，但由于存在未知超参数，我们无法将此方法直接应用于其他GLMs。本研究通过开发适用于特定GLMs类别的新推断方法解决了这一局限。我们的方法基于高维渐近正态性的调整，其可行性体现在即使存在未知超参数仍可实施。具体而言，我们引入了基于新型凸损失函数的估计量及其关联系统，这些是推断的核心组成部分。随后，我们设计了一种基于矩的方法来估计该方法所需的系统参数。由此，我们在高维体系下构建了GLMs的置信区间。我们证明了所提方法具有理想的理论性质，例如强相合性与精确覆盖概率。最后，我们通过实验验证了其有效性。