Cramér type moderate deviations for Studentized U-statistics
ESAIM: Probability and Statistics, Tome 15 (2011), pp. 168-179

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.

DOI : 10.1051/ps/2009014
Classification : 60F10, 60F05
Keywords: moderate deviation, u-statistic, studentized 
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     author = {Lai, Tze Leng and Shao, Qi-Man and Wang, Qiying},
     title = {Cram\'er type moderate deviations for {Studentized} {U-statistics}},
     journal = {ESAIM: Probability and Statistics},
     pages = {168--179},
     year = {2011},
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     zbl = {1271.60044},
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     url = {https://www.numdam.org/articles/10.1051/ps/2009014/}
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Lai, Tze Leng; Shao, Qi-Man; Wang, Qiying. Cramér type moderate deviations for Studentized U-statistics. ESAIM: Probability and Statistics, Tome 15 (2011), pp. 168-179. doi: 10.1051/ps/2009014

[1] I.B. Alberink and V. Bentkus, Berry-Esseen bounds for von-Mises and U-statistics. Lith. Math. J. 41 (2001) 1-16. | Zbl | MR

[2] I.B. Alberink and V. Bentkus, Lyapunov type bounds for U-statistics. Theory Probab. Appl. 46 (2002) 571-588. | Zbl | MR

[3] J.N. Arvesen, Jackknifing U-statistics. Ann. Math. Statist. 40 (1969) 2076-2100. | Zbl | MR

[4] Y.V. Borovskikh and N.C. Weber, Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys 43 (2003) 13-37. | Zbl | MR

[5] Y.V. Borovskikh and N.C. Weber, Large deviations of U-statistics I. Lietuvos Matematikos Rinkinys 43 (2003) 294-316. | Zbl | MR

[6] H. Callaert and N. Veraverbeke, The order of the normal approximation for a studentized U-statistics. Ann. Statist. 9 (1981) 194-200. | Zbl | MR

[7] W. Hoeffding, A class of statistics with asymptotically normal distribution. Ann. Math. Statist. 19 (1948) 293-325. | Zbl | MR

[8] 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. | Zbl | MR

[9] 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. | Zbl | MR

[10] V.S. Koroljuk and V. Yu. Borovskich, Theory of U-statistics. Kluwer Academic Publishers, Dordrecht (1994). | Zbl | MR

[11] Q.M. Shao, Self-normalized large deviations. Ann. Probab. 25 (1997) 285-328. | Zbl | MR

[12] Q.M. Shao, Cramér-type large deviation for Student's t statistic. J. Theorect. Probab. 12 (1999) 387-398. | Zbl | MR

[13] 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. | Zbl | MR

[14] M. Vardemaele and N. Veraverbeke, Cramer type large deviations for studentized U-statistics. Metrika 32 (1985) 165-180. | Zbl | MR

[15] Q. Wang, Bernstein type inequalities for degenerate U-statistics with applications. Ann. Math. Ser. B 19 (1998) 157-166. | Zbl | MR

[16] Q. Wang, B.-Y. Jing and L. Zhao, The Berry-Esséen bound for studentized statistics. Ann. Probab. 28 (2000) 511-535. | Zbl | MR

[17] Q. Wang and N.C. Weber, Exact convergence rate and leading term in the central limit theorem for U-statistics. Statist. Sinica 16 (2006) 1409-1422. | Zbl | MR

[18] L. Zhao, The rate of the normal approximation for a studentized U-statistic. Science Exploration 3 (1983) 45-52. | Zbl | MR

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