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A Bernstein-type inequality for U-statistics and U-processes

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  • Arcones, Miguel A.

Abstract

A Bernstein-type inequality for non-degenerated U-statistics is presented. As the Bernstein inequality for sums of independent identically distributed random variables, in the limit, its tail has the same order as the tail of the limit. We also consider the case of U-processes indexed by a uniformly bounded VC subgraph class of functions. This is applied to obtain exponential inequalities for the local oscillations of the empirical distribution function of U-statistical structure.

Suggested Citation

  • Arcones, Miguel A., 1995. "A Bernstein-type inequality for U-statistics and U-processes," Statistics & Probability Letters, Elsevier, vol. 22(3), pages 239-247, February.
    • Handle: RePEc:eee:stapro:v:22:y:1995:i:3:p:239-247
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    Cited by:

    1. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    2. Sadikoglu, Serhan, 2019. "Essays in econometric theory," Other publications TiSEM 99d83644-f9dc-49e3-a4e1-5, Tilburg University, School of Economics and Management.
    3. Cizek, Pavel & Sadikoglu, Serhan, 2022. "Nonseparable Panel Models with Index Structure and Correlated Random Effects," Other publications TiSEM 7899deb9-0eda-47e6-a3b8-2, Tilburg University, School of Economics and Management.
    4. Eichelsbacher, Peter, 1998. "Moderate and large deviations for U-processes," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 273-296, June.
    5. Miruna Oprescu & Vasilis Syrgkanis & Zhiwei Steven Wu, 2018. "Orthogonal Random Forest for Causal Inference," Papers 1806.03467, arXiv.org, revised Sep 2019.
    6. Krebs, Johannes T.N., 2018. "A large deviation inequality for β-mixing time series and its applications to the functional kernel regression model," Statistics & Probability Letters, Elsevier, vol. 133(C), pages 50-58.
    7. Jianhang Ai & Ondřej Kuželka & Yuyi Wang, 2023. "Hoeffding–Serfling Inequality for U-Statistics Without Replacement," Journal of Theoretical Probability, Springer, vol. 36(1), pages 390-408, March.
    8. Kong, Efang & Linton, Oliver & Xia, Yingcun, 2013. "Global Bahadur Representation For Nonparametric Censored Regression Quantiles And Its Applications," Econometric Theory, Cambridge University Press, vol. 29(5), pages 941-968, October.

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