A third-order bias corrected estimate in generalized linear models

Cordeiro and McCullagh (J Roy Stat Soc Ser B 53:629-643, 1991) derived a second-order bias-corrected estimate, which displays smaller bias than the classical maximum likelihood estimate in generalized linear models. This estimate, although consistent, can display pronounced bias in small to moderately large samples, as shown by Monte Carlo simulations here. In this paper, we obtain a simple matrix formula for the bias of order n(-2) of this estimate, where n is the sample size, and define a third-order bias-corrected estimate in this class of models, which displays much smaller bias in small samples. In particular, some Monte Carlo simulations show that our new estimate can deliver substantial improvements in terms of bias and mean squared errors over the usual maximum likelihood estimate and Cordeiro and McCullagh's estimate.