One-step closed-form estimator for generalized linear model with categorical explanatory variables

The parameters of generalized linear models are generally estimated by the maximum likelihood estimator (MLE), computed using a Newton–Raphson type algorithm that can be time-consuming for a large number of variables or modalities, or a large sample size. Explicit estimators exist for these models but they are not always asymptotically efficient, especially for simple effects models, although they are fast to calculate compared to the MLE. The article proposes a fast and asymptotically efficient estimation of the parameters of generalized linear models with categorical explanatory variables. It is based on a one-step procedure where a single step of the gradient descent is performed on the log-likelihood function initialized from the explicit estimators. This work presents the theoretical results obtained, the simulations carried out and an application to car insurance pricing.