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

TL;DR

This paper establishes optimal-order Berry--Esseen bounds for the convergence to normality of multivariate nonlinear statistics, applying advanced Stein-type methods to various classical and modern statistical estimators.

Contribution

It provides the first known uniform and nonuniform Berry--Esseen bounds for a broad class of multivariate nonlinear statistics, including MLEs and test statistics, with explicit constants.

Findings

Derived optimal-order Berry--Esseen bounds for multivariate statistics

Applied bounds to Pearson's, Student's, Hotelling's, and MLEs

Provided explicit constants for the bounds

Abstract

Uniform and nonuniform Berry--Esseen (BE) bounds of optimal orders on the closeness to normality for general abstract nonlinear statistics are given, which are then used to obtain optimal bounds on the rate of convergence in the delta method for vector statistics. Specific applications to Pearson's, non-central 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. When applied to the well-studied case of the central Student statistic, our general results compare well with known ones in that case, obtained previously by specialized methods. The proofs use a Stein-type method developed by Chen and Shao, a Cram\'er-type of tilt transform, exponential and Rosenthal-type inequalities for sums of random vectors established by Pinelis, Sakhanenko, and Utev, as well as a number of other, quite recent results motivated by this study. The method allows one to obtain bounds with explicit and rather moderate-size constants, at least as far as the uniform bounds are concerned. 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 $\mathbb{E} Y_1=\mathbb{E} Z_1=\mathbb{E} Y_1Z_1=0$ and $\mathbb{E} Y_1^2=\mathbb{E} Z_1^2=1$, where $\sigma:=\sqrt{\mathbb{E} Y_1^2Z_1^2}$.