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

TL;DR

This paper derives explicit bounds on how close the distribution of the MLE is to its normal limit, using advanced probabilistic methods, especially for exponential family models.

Contribution

It provides sharp, explicit upper bounds on the distance between the MLE's distribution and its asymptotic normal distribution, combining the Delta method and Stein's method.

Findings

Sharp bounds on Zolotarev-type distances for MLEs

Applicable to broad exponential family distributions

Enhanced understanding of MLE asymptotic normality

Abstract

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.