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A new large deviation inequality for \(U\)-statistics of order \(2\). (English) Zbl 0957.60031

The second-order \(U\)-statistics \(U_n(f)=\sum_{1\leq i \neq j \leq n} f(\xi_i,\xi_j)\) are considered. Let \(U_n(f-E f)=2(n-1)U_n^{(1)}+U_n^{(2)}\) be the decomposition of \(U_n\) to the linear and second-order canonical projections. The upper bound for the moments \(E (U_n{(2)})^{2k}\) is proved. As a corollary the exponential inequality \[ P(U_n^{(2)}\geq C(f,F,n)x)\leq \exp (6-2\sqrt{x}) \] is derived. This inequality is compared with similar inequalities of M. A. Arcones and E. Giné [Ann. Probab. 21, No. 3, 1494-1542 (1993; Zbl 0789.60031)].

MSC:

60F10 Large deviations

Citations:

Zbl 0789.60031
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References:

[1] M.A. Arcones and E. Giné, Limit Theorems for U-processes. Ann. Probab. 21 ( 1993) 1494-1542. Zbl0789.60031 MR1235426 · Zbl 0789.60031 · doi:10.1214/aop/1176989128
[2] V. De la Peña, Decoupling and Khintchine’s inequalities for U-statistics. Ann. Probab. 20 ( 1992) 1887-1892. Zbl0761.60014 · Zbl 0761.60014 · doi:10.1214/aop/1176989533
[3] B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist. 24 ( 1996) 659-681. Zbl0859.62038 MR1394981 · Zbl 0859.62038 · doi:10.1214/aos/1032894458
[4] B. Laurent and P. Massart, Adaptative estimation of a quadratic functional by model selection ( 1998) preprint. Zbl1105.62328 MR1805785 · Zbl 1105.62328 · doi:10.1214/aos/1015957395
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