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2016 Optimal-order bounds on the rate of convergence to normality in the multivariate delta method
Iosif Pinelis, Raymond Molzon
Electron. J. Statist. 10(1): 1001-1063 (2016). DOI: 10.1214/16-EJS1133


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 $\sigma:=\sqrt{\mathbb{E}Y_{1}^{2}Z_{1}^{2}}$.


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Iosif Pinelis. Raymond Molzon. "Optimal-order bounds on the rate of convergence to normality in the multivariate delta method." Electron. J. Statist. 10 (1) 1001 - 1063, 2016.


Received: 1 August 2015; Published: 2016
First available in Project Euclid: 12 April 2016

zbMATH: 1337.60024
MathSciNet: MR3486424
Digital Object Identifier: 10.1214/16-EJS1133

Primary: 60E15, 60F05, 62F12
Secondary: 60E10, 62F03, 62F05, 62G10, 62G20

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society


Vol.10 • No. 1 • 2016
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