Ryan Godwin and I have a new paper - "Improved Analytic Bias Correction for Maximum Likelihood Estimators". You can download it from here. (This is a revised version, 19 July 2017.)
This paper proposes a modification of the Cox-Snell/Cordeiro-Klein bias correction technique that we've used in our earlier research (including work with Helen Feng and Jacob Schwartz). For some more information about that work, see this earlier post.