Bounds for the normal approximation of the maximum likelihood estimator

TL;DR

This paper provides explicit bounds on how closely the distribution of the maximum likelihood estimator approximates a normal distribution, using Stein's method, for various types of data and models.

Contribution

It introduces a method to derive explicit bounds for the normal approximation of the MLE's distribution without requiring a closed-form expression of the MLE.

Findings

Bounds are applicable to both discrete and continuous distributions.

The approach covers exponential and non-exponential families.

Perturbation method handles boundary cases of the MLE.

Abstract

While the asymptotic normality of the maximum likelihood estimator under regularity conditions is long established, this paper derives explicit bounds for the bounded Wasserstein distance between the distribution of the maximum likelihood estimator (MLE) and the normal distribution. For this task, we employ Stein's method. We focus on independent and identically distributed random variables, covering both discrete and continuous distributions as well as exponential and non-exponential families. In particular, a closed form expression of the MLE is not required. We also use a perturbation method to treat cases where the MLE has positive probability of being on the boundary of the parameter space.