On normal approximations to U-statistics

On normal approximations to U-statistics

Vidmantas Bentkus, Bing-Yi Jing, and Wang Zhou

Source: Ann. Probab. Volume 37, Number 6 (2009), 2174-2199.

Abstract

Let X₁, …, X_n be i.i.d. random observations. Let ${\mathbb{S}=\mathbb{L}+\mathbb{T}}$ be a U-statistic of order k≥2 where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb {T}$ is a stochastically smaller statistic. We show that the rate of convergence to normality for $\mathbb{S}$ can be simply expressed as the rate of convergence to normality for the linear part $\mathbb{L}$ plus a correction term, $(\operatorname{var}\mathbb{T})\ln^{2}(\operatorname{var}\mathbb{T})$ , under the condition ${\mathbb{E}\mathbb{T}^{2}\textless \infty}$ . An optimal bound without this log factor is obtained under a lower moment assumption ${\mathbb{E}|\mathbb{T}|^{\alpha}\textless \infty}$ for ${\alpha \textless 2}$ . Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.

First Page: Show

Primary Subjects: 62E20

Keywords: U-statistics; Berry–Esseen bound; rate of convergence; central limit theorem; normal approximations; self-normalized; Studentized U-statistics

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1258380786
Digital Object Identifier: doi:10.1214/09-AOP474
Mathematical Reviews number (MathSciNet): MR2573555
Zentralblatt MATH identifier: 1186.62025

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