MLE's bias pathology, Fisher's Specification Problem and Model Corrected Maximum Likelihood Estimates (MCMLE)

MLE's inherent bias pathology that is confirmed herein for models with unknown parameters $\theta,\psi$ and MLE $\hat \psi$ function of MLE $\hat \theta$ and the need for model accuracy in Fisher's specification problem suggest updating the Likelihood Equation to be solved for $\psi$ using the model of the data Y in it with $\hat \theta$ replacing $\theta.$ For several models the so-obtained Model Corrected MLE $\hat \psi_{MC}$ reduces either totally or partially the bias when estimating shape parameters. For the Pareto model in particular, with parameters $\theta$ and $\psi$ both unknown, $\hat \psi_{MC}$ reduces the bias but also the variance of $\hat \psi.$ The results contribute in the explanation of the "difference" that has puzzled R. A. Fisher.