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 upper bounds in cases where the MLE cannot be expressed analytically.
"Assessing the multivariate normal approximation of the maximum likelihood estimator from high-dimensional, heterogeneous data." Electron. J. Statist. 12 (2) 3794 - 3828, 2018. https://doi.org/10.1214/18-EJS1492