On normal approximations to U-statistics
Open Access
- 1 November 2009
- journal article
- research article
- Published by Institute of Mathematical Statistics in The Annals of Probability
- Vol. 37 (6), 2174-2199
- https://doi.org/10.1214/09-aop474
Abstract
Let X-1, ..., X-n be i.i.d. random observations. Let S = L + T be a U-statistic of order k >= 2 where L is a linear statistic having asymptotic normal distribution, and T is a stochastically smaller statistic. We show that the rate of convergence to normality for S can be simply expressed as the rate of convergence to normality for the linear part L plus a correction term, (varT) ln(2) (varT), under the condition ET2 < infinity. An optimal bound without this log factor is obtained under a lower moment assumption E vertical bar T vertical bar(alpha) < infinity 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.Keywords
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This publication has 15 references indexed in Scilit:
- Normal approximation for nonlinear statistics using a concentration inequality approachBernoulli, 2007
- Hoeffding-ANOVA decompositions for symmetric statistics of exchangeable observationsThe Annals of Probability, 2004
- Edgeworth expansion for U-statistics under minimal conditionsThe Annals of Statistics, 2003
- Orthogonal decomposition of finite population statistics and its applications to distributional asymptoticsThe Annals of Statistics, 2001
- Lower Estimates of the Convergence Rate for $U$-StatisticsThe Annals of Probability, 1994
- A Berry-Esseen Bound for Functions of Independent Random VariablesThe Annals of Statistics, 1989
- Approximation of Nondegenerate U -StatisticsTheory of Probability and Its Applications, 1986
- A Berry-Esseen bound for symmetric statisticsProbability Theory and Related Fields, 1984
- Applications of Anova Type Decompositions for Comparisons of Conditional Variance Statistics Including Jackknife EstimatesThe Annals of Statistics, 1982
- A Class of Statistics with Asymptotically Normal DistributionThe Annals of Mathematical Statistics, 1948