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Cramér type moderate deviations for Studentized U-statistics******

Published online by Cambridge University Press:  05 January 2012

Tze Leng Lai ,
Qi-Man Shao  and
Qiying Wang
Tze Leng Lai
Affiliation:
Department of Statistics, Stanford University Stanford, CA CA 94305-4065, USA; lait@stanford.edu
Qi-Man Shao
Affiliation:
Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon Hong Kong, P.R. China; maqmshao@ust.hk
Qiying Wang
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia; qiying@maths.usyd.edu.au
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Abstract

Let Tn be a Studentized U-statistic. It is proved that a Cramér typemoderate deviation P(Tn ≥ x)/(1 − Φ(x)) → 1 holds uniformly in x ∈ [0, o(n1/6))when the kernel satisfies some regular conditions.

Keywords

Moderate deviationu-statisticstudentized

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 15: Supplement: In honor of Marc Yor , 2011 , pp. 168 - 179
Copyright
© EDP Sciences, SMAI, 2011

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References

Alberink, I.B. and Bentkus, V., Berry-Esseen bounds for von-Mises and U-statistics. Lith. Math. J. 41 (2001) 1–16. CrossRef
Alberink, I.B. and Bentkus, V., Lyapunov type bounds for U-statistics. Theory Probab. Appl. 46 (2002) 571–588. CrossRef
Arvesen, J.N., Jackknifing U-statistics. Ann. Math. Statist. 40 (1969) 2076–2100. CrossRef
Borovskikh, Y.V. and Weber, N.C., Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys 43 (2003) 13–37.
Borovskikh, Y.V. and Weber, N.C., Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys 43 (2003) 294–316.
Callaert, H. and Veraverbeke, N., The order of the normal approximation for a studentized U-statistics. Ann. Statist. 9 (1981) 194–200. CrossRef
Hoeffding, W., A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 (1948) 293–325. CrossRef
Jing, B.-Y., Shao, Q.M. and Wang, Q., Self-normalized Cramér-type large deviation for independent random variables. Ann. Probab. 31 (2003) 2167–2215.
Jing, B.-Y., Shao, Q.M., Zhou, W., Saddlepoint approximation for Student's t-statistic with no moment conditions. Ann. Statist. 32 (2004) 2679–2711. CrossRef
V.S. Koroljuk and V. Yu. Borovskich, Theory of U-statistics. Kluwer Academic Publishers, Dordrecht (1994).
Shao, Q.M., Self-normalized large deviations. Ann. Probab. 25 (1997) 285–328.
Shao, Q.M., Cramér-type large deviation for Student's t statistic. J. Theorect. Probab. 12 (1999) 387–398.
De La Pena, V.H., Klass, M.J. and Lai, T.L., Self-normalized processes: exponential inequalities, moment bound and iterated logarithm laws. Ann. Probab. 32 (2004) 1902–1933.
Vardemaele, M. and Veraverbeke, N., Cramer type large deviations for studentized U-statistics. Metrika 32 (1985) 165–180. CrossRef
Wang, Q., Bernstein type inequalities for degenerate U-statistics with applications. Ann. Math. Ser. B 19 (1998) 157–166.
Wang, Q., Jing, B.-Y. and Zhao, L., The Berry-Esséen bound for studentized statistics. Ann. Probab. 28 (2000) 511–535. CrossRef
Wang, Q. and Weber, N.C., Exact convergence rate and leading term in the central limit theorem for U-statistics. Statist. Sinica 16 (2006) 1409–1422.
Zhao, L., The rate of the normal approximation for a studentized U-statistic. Science Exploration 3 (1983) 45–52.