Bias reduction of maximum likelihood estimates

It is shown how, in regular parametric problems, the first-order term is removed from the asymptotic bias of maximum likelihood estimates by a suitable modification of the score function. In exponential families with canonical parameterization the effect is to penalize the likelihood by the Jeffreys invariant prior. In binomial logistic models, Poisson log linear models and certain other generalized linear models, the Jeffreys prior penalty function can be imposed in standard regression software using a scheme of iterative adjustments to the data.