  The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a
cornerstone of statistical theory. In the present paper, we provide sharp
explicit upper bounds on Zolotarev-type distances between the exact, unknown
distribution of the MLE and its limiting normal distribution. Our approach to
this fundamental issue is based on a sound combination of the Delta method,
Stein's method, Taylor expansions and conditional expectations, for the
classical situations where the MLE can be expressed as a function of a sum of
independent and identically distributed terms. This encompasses in particular
the broad exponential family of distributions.
