## Electronic Journal of Statistics

### Optimal-order bounds on the rate of convergence to normality in the multivariate delta method

#### Abstract

Uniform and nonuniform Berry–Esseen (BE) bounds of optimal orders on the rate of convergence to normality in the delta method for vector statistics are obtained. The results are applicable almost as widely as the delta method itself – except that, quite naturally, the order of the moments needed to be finite is generally $3/2$ times as large as that for the corresponding central limit theorems. Our BE bounds appear new even for the one-dimensional delta method, that is, for smooth functions of the sample mean of univariate random variables. Specific applications to Pearson’s, noncentral Student’s and Hotelling’s statistics, sphericity test statistics, a regularized canonical correlation, and maximum likelihood estimators (MLEs) are given; all these uniform and nonuniform BE bounds appear to be the first known results of these kinds, except for uniform BE bounds for MLEs. The new method allows one to obtain bounds with explicit and rather moderate-size constants. For instance, one has the uniform BE bound $3.61\mathbb{E}(Y_{1}^{6}+Z_{1}^{6})\,(1+\sigma^{-3})/\sqrt{n}$ for the Pearson sample correlation coefficient based on independent identically distributed random pairs $(Y_{1},Z_{1}),\dots,(Y_{n},Z_{n})$ with $\mathbb{E} Y_{1}=\mathbb{E}Z_{1}=\mathbb{E}Y_{1}Z_{1}=0$ and $\mathbb{E}Y_{1}^{2}=\mathbb{E}Z_{1}^{2}=1$, where $\sigma:=\sqrt{\mathbb{E}Y_{1}^{2}Z_{1}^{2}}$.

#### Article information

Source
Electron. J. Statist., Volume 10, Number 1 (2016), 1001-1063.

Dates
First available in Project Euclid: 12 April 2016

https://projecteuclid.org/euclid.ejs/1460463653

Digital Object Identifier
doi:10.1214/16-EJS1133

Mathematical Reviews number (MathSciNet)
MR3486424

Zentralblatt MATH identifier
1337.60024

#### Citation

Pinelis, Iosif; Molzon, Raymond. Optimal-order bounds on the rate of convergence to normality in the multivariate delta method. Electron. J. Statist. 10 (2016), no. 1, 1001--1063. doi:10.1214/16-EJS1133. https://projecteuclid.org/euclid.ejs/1460463653