On normal approximations to $U$-statistics

Abstract

Let $X_1,...,X_n$ be i.i.d. random observations. Let $\mathbb{S}=\mathbb{L}+\mathbb{T}$ be a $U$-statistic of order $k\ge 2$ 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, $(\mathrm{var}\mathbb{T}){\ln }^2(\mathrm{var}\mathbb{T})$, under the condition $\mathbb{E}{\mathbb{T}}^2<\infty$. An optimal bound without this $\log$ factor is obtained under a lower moment assumption $\mathbb{E}|\mathbb{T}{|}^{\alpha }<\infty$ for $\alpha <2$. 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.

Published in at http://dx.doi.org/10.1214/09-AOP474 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Normal approximation for nonlinear statistics using a concentration inequality approach
• On normal approximations to $U$-statistics
• Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observations

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• Second Order Approximations for Slightly Trimmed Sums
• Rates of convergence in the CLT for nonlinear statistics under relaxed moment conditions