Optimal-order bounds on the rate of convergence to normality for maximum likelihood estimators

Understand

It is well known that under general regularity conditions the distribution of the maximum likelihood estimator (MLE) is asymptotically normal.

• Very recently, bounds of the optimal order $O(1/\sqrt n)$ on the closeness of the distribution of the MLE to normality in the so-called bounded Wasserstein distance were obtained, where $n$ is the sample size.
• However, the corresponding bounds on the Kolmogorov distance were only of the order $O(1/n^{1/4})$.
• In this note, bounds of the optimal order $O(1/\sqrt n)$ on the closeness of the distribution of the MLE to normality in the Kolmogorov distance are given, as well as their nonuniform counterparts, which work better for large deviations of the MLE.