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Robust estimation of U-statistics

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  • Joly, Emilien
  • Lugosi, Gábor

Abstract

An important part of the legacy of Evarist Giné is his fundamental contributions to our understanding of U-statistics and U-processes. In this paper we discuss the estimation of the mean of multivariate functions in case of possibly heavy-tailed distributions. In such situations, reliable estimates of the mean cannot be obtained by usual U-statistics. We introduce a new estimator, based on the so-called median-of-means technique. We develop performance bounds for this new estimator that generalizes an estimate of Arcones and Giné (1993), showing that the new estimator performs, under minimal moment conditions, as well as classical U-statistics for bounded random variables. We discuss an application of this estimator to clustering.

Suggested Citation

  • Joly, Emilien & Lugosi, Gábor, 2016. "Robust estimation of U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3760-3773.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:12:p:3760-3773
    DOI: 10.1016/j.spa.2016.04.021
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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. Salim Bouzebda & Thouria El-hadjali & Anouar Abdeldjaoued Ferfache, 2023. "Uniform in Bandwidth Consistency of Conditional U-statistics Adaptive to Intrinsic Dimension in Presence of Censored Data," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1548-1606, August.
    3. Salim Bouzebda & Inass Soukarieh, 2022. "Non-Parametric Conditional U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design," Mathematics, MDPI, vol. 11(1), pages 1-69, December.
    4. Salim Bouzebda, 2024. "Limit Theorems in the Nonparametric Conditional Single-Index U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design," Mathematics, MDPI, vol. 12(13), pages 1-81, June.
    5. Inass Soukarieh & Salim Bouzebda, 2022. "Exchangeably Weighted Bootstraps of General Markov U -Process," Mathematics, MDPI, vol. 10(20), pages 1-42, October.

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