Bretagnolle, Jean A new large deviation inequality for \(U\)-statistics of order \(2\). (English) Zbl 0957.60031 ESAIM, Probab. Stat. 3, 151-162 (1999). 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)]. Reviewer: Aleksandras Plikusas (Vilnius) Cited in 3 Documents MSC: 60F10 Large deviations Keywords:\(U\)-statistics; exponential inequalities Citations:Zbl 0789.60031 PDF BibTeX XML Cite \textit{J. Bretagnolle}, ESAIM, Probab. Stat. 3, 151--162 (1999; Zbl 0957.60031) Full Text: DOI EuDML OpenURL 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 [2] V. De la Peña, Decoupling and Khintchine’s inequalities for U-statistics. Ann. Probab. 20 ( 1992) 1887-1892. Zbl0761.60014 · Zbl 0761.60014 [3] B. Laurent, Efficient estimation of integral functionals of a density. Ann. Statist. 24 ( 1996) 659-681. Zbl0859.62038 MR1394981 · Zbl 0859.62038 [4] B. Laurent and P. Massart, Adaptative estimation of a quadratic functional by model selection ( 1998) preprint. Zbl1105.62328 MR1805785 · Zbl 1105.62328 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.