Electronic Journal of Statistics

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

Iosif Pinelis and Raymond Molzon

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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

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

Received: August 2015
First available in Project Euclid: 12 April 2016

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60E15: Inequalities; stochastic orderings 62F12: Asymptotic properties of estimators
Secondary: 60E10: Characteristic functions; other transforms 62F03: Hypothesis testing 62F05: Asymptotic properties of tests 62G10: Hypothesis testing 62G20: Asymptotic properties

Berry–Esseen bound canonical correlation delta method rates of convergence Cramér’s tilt exponential inequalities maximum likelihood estimators noncentral Hotelling’s statistic noncentral Student’s statistic nonlinear statistics Pearson’s correlation coefficient sphericity test


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

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