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A class of U-statistics and asymptotic normality of the number of k-clusters

Author

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  • Bhattacharya, Rabi N.
  • Ghosh, Jayanta K.

Abstract

A central limit theorem is proved for a class of U-statistics whose kernel depends on the sample size and for which the projection method may fail, since several terms in the Hoeffding decomposition contribute to the limiting variance. As an application we derive the asymptotic normality of the number of Poisson k-clusters in a cube of increasing size in Rd. We also extend earlier results of Jammalamadaka and Janson to general kernels and to general orders k > 2 of the kernel.

Suggested Citation

  • Bhattacharya, Rabi N. & Ghosh, Jayanta K., 1992. "A class of U-statistics and asymptotic normality of the number of k-clusters," Journal of Multivariate Analysis, Elsevier, vol. 43(2), pages 300-330, November.
    • Handle: RePEc:eee:jmvana:v:43:y:1992:i:2:p:300-330
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    Cited by:

    1. Lin Liu & Chang Li, 2023. "New $\sqrt{n}$-consistent, numerically stable higher-order influence function estimators," Papers 2302.08097, arXiv.org.
    2. Lachièze-Rey, Raphaël & Peccati, Giovanni, 2013. "Fine Gaussian fluctuations on the Poisson space II: Rescaled kernels, marked processes and geometric U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(12), pages 4186-4218.
    3. Bock, Hans H., 1996. "Probabilistic models in cluster analysis," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 5-28, November.
    4. Robins, James M. & Li, Lingling & Tchetgen, Eric Tchetgen & van der Vaart, Aad, 2016. "Asymptotic normality of quadratic estimators," Stochastic Processes and their Applications, Elsevier, vol. 126(12), pages 3733-3759.

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