  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.
