Covariance matrix of the bias-corrected maximum likelihood estimator in generalized linear models

For the first time, we obtain a general formula for the $$n^{-2}$$ asymptotic covariance matrix of the bias-corrected maximum likelihood estimators of the linear parameters in generalized linear models, where $$n$$ is the sample size. The usefulness of the formula is illustrated in order to obtain a better estimate of the covariance of the maximum likelihood estimators and to construct better Wald statistics. Simulation studies and an application support our theoretical results.