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

Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method