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

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 result is tailored for the broad class of one-parameter exponential family distributions. A great part of this work has been completed while Andreas Anastasiou was working at the University of Oxford. Andreas Anastasiou was supported by a Teaching Assistantship Bursary from the Department of Statistics, University of Oxford, and EPSRC grant EP/K503113/1. Christophe Ley thanks the Fonds National de la Recherche Scientifique, Communauté Fran¸caise de Belgique, for financial support via a Mandat de Chargé de Recherche FNRS..