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

By Iosif Pinelis, Raymond Molzon. 2009

51 citations recorded by the source corpus.

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 $\sigma:=\sqrt{\mathbb{E} Y_1^2Z_1^2}$.

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Byzantine-Robust Distributed Learning: Towards Optimal Statistical Rates (2018, methodology): “Instead, we apply the following bound in [6] on functions of sample means.”
Bootstrapping and sample splitting for high-dimensional, assumption-lean inference (2016, background): “Existing Berry-Esseen results for nonlinear parameters are given in Pinelis and Molzon (2016); Shao et al. (2016); Chen and Shao (2007); Anastasiou and Reinert (2014); Anastasiou and Ley (2015); Anastasiou and Gaunt (2016).”