Multivariate normal approximation of the maximum likelihood estimator via the delta method

TL;DR

This paper develops explicit bounds on how closely the maximum likelihood estimator approximates a multivariate normal distribution, using the delta and Stein's methods, especially for estimators based on sums of i.i.d. vectors.

Contribution

It introduces a novel approach combining the delta method and Stein's method to derive explicit bounds for the distributional approximation of the MLE in multivariate settings.

Findings

Provides explicit upper bounds for distributional distance between MLE and normal distribution.

Applies the bounds to the MLE of normal distribution parameters, demonstrating practical utility.

Abstract

We use the delta method and Stein's method to derive, under regularity conditions, explicit upper bounds for the distributional distance between the distribution of the maximum likelihood estimator (MLE) of a $d$-dimensional parameter and its asymptotic multivariate normal distribution. Our bounds apply in situations in which the MLE can be written as a function of a sum of i.i.d. $t$-dimensional random vectors. We apply our general bound to establish a bound for the multivariate normal approximation of the MLE of the normal distribution with unknown mean and variance.