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On the law of the iterated logarithm for canonical U-statistics and processes

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  • Arcones, Miguel A.
  • Giné, Evarist

Abstract

The law of the iterated logarithm for canonical or completely degenerate U-statistics with square integrable kernel h is proved, for h taking values in 1, 7 and, in general, in a type 2 separable Banach space. The LIL is also obtained for U-processes indexed by canonical Vapnik-Cervonenkis classes of functions with square integrable envelope and, in this regard, an equicontinuity condition equivalent to the LIL property is quite helpful. Some of these results are then applied to obtain the a.s. exact order of the remainder term in the linearization of the product limit estimator for truncated data; a consequence for density estimation is also included.

Suggested Citation

  • Arcones, Miguel A. & Giné, Evarist, 1995. "On the law of the iterated logarithm for canonical U-statistics and processes," Stochastic Processes and their Applications, Elsevier, vol. 58(2), pages 217-245, August.
    • Handle: RePEc:eee:spapps:v:58:y:1995:i:2:p:217-245
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    References listed on IDEAS

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    1. Alexander, Kenneth S. & Talagrand, Michel, 1989. "The law of the iterated logarithm for empirical processes on Vapnik-Cervonenkis classes," Journal of Multivariate Analysis, Elsevier, vol. 30(1), pages 155-166, July.
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    6. Arcones, Miguel A. & Giné, Evarist, 1994. "U-processes indexed by Vapnik-Cervonenkis classes of functions with applications to asymptotics and bootstrap of U-statistics with estimated parameters," Stochastic Processes and their Applications, Elsevier, vol. 52(1), pages 17-38, August.
    7. Dehling, Herold, 1989. "Complete convergence of triangular arrays and the law of the iterated logarithm for U-statistics," Statistics & Probability Letters, Elsevier, vol. 7(4), pages 319-321, February.
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    1. C. Sánchez-Sellero & W. González-Manteiga & R. Cao, 1999. "Bandwidth Selection in Density Estimation with Truncated and Censored Data," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 51(1), pages 51-70, March.
    2. Sun, Liuquan & Zhou, Yong, 1998. "Sequential confidence bands for densities under truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 40(1), pages 31-41, September.
    3. 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.
    4. Jon A. Wellner, 2017. "The Bennett-Orlicz Norm," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(2), pages 355-383, August.
    5. Subramanian, Sundarraman & Bandyopadhyay, Dipankar, 2008. "Semiparametric left truncation and right censorship models with missing censoring indicators," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2572-2577, November.
    6. Sun, Liuquan & Zhou, Xian, 2001. "Survival function and density estimation for truncated dependent data," Statistics & Probability Letters, Elsevier, vol. 52(1), pages 47-57, March.
    7. Zhou, Yong & Yip, Paul S. F., 1999. "A Strong Representation of the Product-Limit Estimator for Left Truncated and Right Censored Data," Journal of Multivariate Analysis, Elsevier, vol. 69(2), pages 261-280, May.
    8. Eichelsbacher, Peter, 2000. "Moderate deviations for degenerate U-processes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 255-279, June.
    9. Sun, Liuquan & Zhu, Lixing, 2000. "A semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 217-227, July.
    10. Radosław Adamczak & Rafał Latała, 2008. "The LIL for U-Statistics in Hilbert Spaces," Journal of Theoretical Probability, Springer, vol. 21(3), pages 704-744, September.
    11. Mikosch, T. & Norvaisa, R., 1997. "Uniform convergence of the empirical spectral distribution function," Stochastic Processes and their Applications, Elsevier, vol. 70(1), pages 85-114, October.
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