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Title: On normal approximations to U -statistics
Authors: Bentkus, V.
Jing, B.-Y.
Zhou, W. 
Keywords: Berry esseen bound
Central limit theorem
Normal approximations
Rate of convergence
Studentized U -statistics
Issue Date: Nov-2009
Citation: Bentkus, V., Jing, B.-Y., Zhou, W. (2009-11). On normal approximations to U -statistics. Annals of Probability 37 (6) : 2174-2199. ScholarBank@NUS Repository.
Abstract: Let X1,..., Xn be i.i.d. random observations. Let S{double-struck} = L{double-struck} + T{double-struck} be a U -statistic of order k ≥ 2 where L{double-struck} is a linear statistic having asymptotic normal distribution, and T{double-struck} is a stochastically smaller statistic. We show that the rate of convergence to normality for S{double-struck} can be simply expressed as the rate of convergence to normality for the linear part L{double-struck} plus a correction term, (var T{double-struck}) ln2(var T{double-struck}), under the condition E{double-struck}T{double-struck}2 < ∞. An optimal bound without this log factor is obtained under a lower moment assumption E{double-struck}|T{double-struck}|α < ∞ for α
Source Title: Annals of Probability
ISSN: 00911798
DOI: 10.1214/09-AOP474
Appears in Collections:Staff Publications

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