  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.
These results are based on previously obtained general optimal-order bounds on
the rate of convergence to normality in the multivariate delta method. The
crucial observation is that, under natural conditions, the MLE can be tightly
enough bracketed between two smooth enough functions of the sum of independent
random vectors, which makes the delta method applicable.
