Assessing the multivariate normal approximation of the maximum likelihood estimator from high-dimensional, heterogeneous data

TL;DR

This paper provides explicit bounds on how closely the maximum likelihood estimator approximates a multivariate normal distribution in high-dimensional, heterogeneous data settings, extending classical asymptotic results.

Contribution

It introduces explicit upper bounds for the distributional distance between the MLE and the normal distribution in high-dimensional, non-i.i.d. data, even when the MLE lacks a closed form.

Findings

Explicit bounds are derived for the distributional distance.

Results apply to high-dimensional, heterogeneous data.

Bounds are valid even without an analytical form of the MLE.

Abstract

The asymptotic normality of the maximum likelihood estimator (MLE) under regularity conditions is a cornerstone of statistical theory. In this paper, we give explicit upper bounds on the distributional distance between the distribution of the MLE of a vector parameter, and the multivariate normal distribution. We work with possibly high-dimensional, independent but not necessarily identically distributed random vectors. In addition, we obtain explicit upper bounds even in cases where the MLE cannot be expressed analytically.