is always a positive probability of data separation [@albert+anderson:1984] that results in infinite maximum likelihood estimates. Then, asymptotic bias correction, bootstrap, iterated bootstrap, and jackknife cannot be applied. The direct dependence on $\hat{{}\theta}$ may be more consequential for naive implementations of the latter three methods because they are simulation-based; even if data separation did not occur for the original sample, there is always positive probability that it will occur for at least one of the simulated samples. There is no easy way of knowing this before carrying out the simulation and such cases can only be handled in an ad-hoc way.

The current work develops a novel method for the reduction of the asymptotic bias of $M$-estimators from general, unbiased estimating functions. We call the new estimation method reduced-bias $M$-estimation, or RB$M$-estimation in short. Like the adjusted scores approach in @firth:1993, the new method relies on additive adjustments to the unbiased estimating functions that are bounded in probability, and results in estimators with bias of lower asymptotic order than the original $M$-estimators. The key difference is that the empirical adjustments introduced here depend only on the first two derivatives of the contributions to the estimating functions, and they require neither 