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

Tze Leng Lai; Qi-Man Shao; Qiying Wang

ESAIM: Probability and Statistics (2012)

- Volume: 15, page 168-179
- ISSN: 1292-8100

## Access Full Article

top## Abstract

top## How to cite

topLai, Tze Leng, Shao, Qi-Man, and Wang, Qiying. "Cramér type moderate deviations for Studentized U-statistics******." ESAIM: Probability and Statistics 15 (2012): 168-179. <http://eudml.org/doc/222449>.

@article{Lai2012,

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

author = {Lai, Tze Leng, Shao, Qi-Man, Wang, Qiying},

journal = {ESAIM: Probability and Statistics},

keywords = {Moderate deviation; u-statistic; studentized; moderate deviation; -statistic; Studentized statistic},

language = {eng},

month = {1},

pages = {168-179},

publisher = {EDP Sciences},

title = {Cramér type moderate deviations for Studentized U-statistics******},

url = {http://eudml.org/doc/222449},

volume = {15},

year = {2012},

}

TY - JOUR

AU - Lai, Tze Leng

AU - Shao, Qi-Man

AU - Wang, Qiying

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

JO - ESAIM: Probability and Statistics

DA - 2012/1//

PB - EDP Sciences

VL - 15

SP - 168

EP - 179

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

LA - eng

KW - Moderate deviation; u-statistic; studentized; moderate deviation; -statistic; Studentized statistic

UR - http://eudml.org/doc/222449

ER -

## References

top- I.B. Alberink and V. Bentkus, Berry-Esseen bounds for von-Mises and U-statistics. Lith. Math. J.41 (2001) 1–16.
- I.B. Alberink and V. Bentkus, Lyapunov type bounds for U-statistics. Theory Probab. Appl.46 (2002) 571–588.
- J.N. Arvesen, Jackknifing U-statistics. Ann. Math. Statist.40 (1969) 2076–2100.
- Y.V. Borovskikh and N.C. Weber, Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys43 (2003) 13–37.
- Y.V. Borovskikh and N.C. Weber, Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys43 (2003) 294–316.
- H. Callaert and N. Veraverbeke, The order of the normal approximation for a studentized U-statistics. Ann. Statist.9 (1981) 194–200.
- W. Hoeffding, A class of statistics with asymptotically normal distribution. Ann. Math. Statist.19 (1948) 293–325.
- B.-Y. Jing, Q.M. Shao and Q. Wang, Self-normalized Cramér-type large deviation for independent random variables. Ann. Probab.31 (2003) 2167–2215.
- B.-Y. Jing, Q.M. Shao, W. Zhou, Saddlepoint approximation for Student's t-statistic with no moment conditions. Ann. Statist.32 (2004) 2679–2711.
- V.S. Koroljuk and V. Yu. Borovskich, Theory of U-statistics. Kluwer Academic Publishers, Dordrecht (1994).
- Q.M. Shao, Self-normalized large deviations. Ann. Probab.25 (1997) 285–328.
- Q.M. Shao, Cramér-type large deviation for Student's t statistic. J. Theorect. Probab.12 (1999) 387–398.
- V.H. De La Pena, M.J. Klass and T.L. Lai, Self-normalized processes: exponential inequalities, moment bound and iterated logarithm laws. Ann. Probab.32 (2004) 1902–1933.
- M. Vardemaele and N. Veraverbeke, Cramer type large deviations for studentized U-statistics. Metrika32 (1985) 165–180.
- Q. Wang, Bernstein type inequalities for degenerate U-statistics with applications. Ann. Math. Ser. B19 (1998) 157–166.
- Q. Wang, B.-Y. Jing and L. Zhao, The Berry-Esséen bound for studentized statistics. Ann. Probab.28 (2000) 511–535.
- Q. Wang and N.C. Weber, Exact convergence rate and leading term in the central limit theorem for U-statistics. Statist. Sinica16 (2006) 1409–1422.
- L. Zhao, The rate of the normal approximation for a studentized U-statistic. Science Exploration3 (1983) 45–52.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.