## Electronic Journal of Statistics

### Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators

Iosif Pinelis

#### Abstract

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 [2, 1], 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 paper, 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 in tail zones of the distribution of the MLE. These results are based in part 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. It appears that the nonuniform bounds for MLEs in general have no precedents in the existing literature; a special case was recently treated by Pinelis and Molzon [20]. The results can be extended to $M$-estimators.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 1 (2017), 1160-1179.

Dates
First available in Project Euclid: 11 April 2017

https://projecteuclid.org/euclid.ejs/1491897618

Digital Object Identifier
doi:10.1214/17-EJS1264

Mathematical Reviews number (MathSciNet)
MR3634332

Zentralblatt MATH identifier
1361.62013

#### Citation

Pinelis, Iosif. Optimal-order uniform and nonuniform bounds on the rate of convergence to normality for maximum likelihood estimators. Electron. J. Statist. 11 (2017), no. 1, 1160--1179. doi:10.1214/17-EJS1264. https://projecteuclid.org/euclid.ejs/1491897618

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