MLE’s Bias Pathology, Model Updated MLE, and Wallace’s Minimum Message Length Method

The inherent bias pathology of the maximum likelihood estimation method is confirmed for models with unknown parameters θ and ψ when maximum likelihood estimate (MLE) ψ̂ is function of MLE θ̂. To reduce ψ̂'s bias the likelihood equation to be solved for ψ is updated using the model for the data Y in it. For various models with ψ a shape parameter model updated (MU) MLE, ψ̂
_MU
, reduces ψ̂'s bias. For the Pareto model ψ̂
_MU
reduces in addition ψ̂'s variance. The results explain the difference that puzzled Fisher, between biased ψ̂ and the unbiased estimate he obtained for two models with the abandoned two-stage procedure used in MUMLE's implementation. ψ̂
_MU
is also obtained with the minimum message length method thus motivating the use of priors in frequentist inference.