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Rates of convergence for U-Statistics with varying kernels

Published online by Cambridge University Press:  17 April 2009

N.C. Weber
Affiliation:
Department of Mathematical Statistics, University of Sydney, Sydney, New South Wales 2006, Australia.
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Abstract

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Let Un be a U-statistic whose kernel depends on the size n of the sample under consideration. It is shown that when Un is suitably normalised its distribution function differs in Lp norm from the distribution function of a standard normal variable by a term of O(n).

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1980

References

[1]Callaert, Herman and Janssen, Paul, “The Berry-Esseen theorem for U-statistics”, Ann. Statist. 6 (1978), 417421.Google Scholar
[2]Erickson, R.V., “On an Lp version of the Berry-Esseen theorem for independent and m-dependent variables”, Ann. Prob. 1 (1973), 497503.CrossRefGoogle Scholar
[3]Erickson, R.V. and Koul, H.L., “L1 rates of convergence for linear rank statistics”, Ann. Statist. 4 (1976), 771774.Google Scholar
[4]Kester, Adri, “Asymptotic normality of the number of small distances between random points in a cube”, Stochastic Process. Appl. 3 (1975), 4554.Google Scholar
[5]Silverman, Bernard, Brown, Tim, “Short distances, flat triangles and Poisson limits”, J. Appl. Probab. 15 (1978), 815825.Google Scholar