The widespread use of maximum Jeffreys'-prior penalized likelihood in binomial-response generalized linear models, and in logistic regression, in particular, are supported by the results of Kosmidis and Firth (2021, Biometrika), who show that the resulting estimates are also always finite-valued, even in cases where the maximum likelihood estimates are not, which is a practical issue regardless of the size of the data set. In logistic regression, the implied adjusted score equations are formally bias-reducing in asymptotic frameworks with a fixed number of parameters and appear to deliver a substantial reduction in the persistent bias of the maximum likelihood estimator in high-dimensional settings where the number of parameters grows asymptotically linearly and slower than the number of observations. In this work, we develop and present two new variants of iteratively reweighted least squares for estimating generalized linear models with adjusted score equations for mean bias reduction and maximization of the likelihood penalized by a positive power of the Jeffreys-prior penalty, which eliminate the requirement of storing $O(n)$ quantities in memory, and can operate with data sets that exceed computer memory or even hard drive capacity. We achieve that through incremental QR decompositions, which enable IWLS iterations to have access only to data chunks of predetermined size. We assess the procedures through a real-data application with millions of observations.

翻译：Kosmidis和Firth（2021, Biometrika）的结果支持了最大Jeffreys先验惩罚似然在二项响应广义线性模型（尤其是逻辑回归）中的广泛应用，该研究表明，即使在极大似然估计无法保证有限值的情况下（此问题与数据集规模无关），所得估计量始终为有限值。在逻辑回归中，所隐含的调整得分方程在参数数量固定的渐近框架中具有正式的意义，并且在参数数量随观测数呈渐近线性增长且增速较慢的高维场景下，能显著降低极大似然估计量的持久偏差。本研究开发并提出了两种迭代重加权最小二乘的新变体，用于估计具有均值偏差缩减调整得分方程和Jeffreys先验惩罚正幂极大化似然的广义线性模型。这些方法消除了在内存中存储$O(n)$个量的需求，并能够处理超出计算机内存甚至硬盘容量的数据集。我们通过增量QR分解实现了上述目标，使得IWLS迭代仅需访问预定大小的数据块。通过数百万观测值的真实数据应用，我们评估了这些过程的性能。