IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v58y1995i2p217-245.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(94)00023-M
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    2. Diehl, Sabine & Stute, Winfried, 1988. "Kernel density and hazard function estimation in the presence of censoring," Journal of Multivariate Analysis, Elsevier, vol. 25(2), pages 299-310, May.
    3. Gijbels, I. & Wang, J. L., 1993. "Strong Representations of the Survival Function Estimator for Truncated and Censored Data with Applications," Journal of Multivariate Analysis, Elsevier, vol. 47(2), pages 210-229, November.
    4. Dehling, Herold, 1989. "The functional law of the iterated logarithm for von Mises functionals and multiple Wiener integrals," Journal of Multivariate Analysis, Elsevier, vol. 28(2), pages 177-189, February.
    5. Arcones, M. A., 1993. "The Law of the Iterated Logarithm for U-Processes," Journal of Multivariate Analysis, Elsevier, vol. 47(1), pages 139-151, October.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    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. 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.
    4. Eichelsbacher, Peter, 2000. "Moderate deviations for degenerate U-processes," Stochastic Processes and their Applications, Elsevier, vol. 87(2), pages 255-279, June.
    5. Sun, Liuquan & Zhu, Lixing, 2000. "A semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 217-227, July.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    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.
    12. Wendler, Martin, 2012. "U-processes, U-quantile processes and generalized linear statistics of dependent data," Stochastic Processes and their Applications, Elsevier, vol. 122(3), pages 787-807.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Eichelsbacher, Peter, 1998. "Moderate and large deviations for U-processes," Stochastic Processes and their Applications, Elsevier, vol. 74(2), pages 273-296, June.
    3. 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.
    4. Sun, Liuquan & Zhu, Lixing, 2000. "A semiparametric model for truncated and censored data," Statistics & Probability Letters, Elsevier, vol. 48(3), pages 217-227, July.
    5. Einmahl, J.H.J. & Deheuvels, P., 2000. "Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications," Other publications TiSEM ac9bbdc0-62f8-4b48-9a84-1, Tilburg University, School of Economics and Management.
    6. Junshan Shen & Shuyuan He, 2007. "Empirical likelihood for the difference of quantiles under censorship," Statistical Papers, Springer, vol. 48(3), pages 437-457, September.
    7. Pedro Delicado & Juan Romo, 1998. "Constant coefficient tests for random coefficient regression," Economics Working Papers 329, Department of Economics and Business, Universitat Pompeu Fabra.
    8. Prewitt, Kathryn & Gürler, Ulkü, 1999. "Variance of the bivariate density estimator for left truncated right censored data," Statistics & Probability Letters, Elsevier, vol. 45(4), pages 351-358, December.
    9. Elisa–María Molanes-López & Ricardo Cao, 2008. "Relative density estimation for left truncated and right censored data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(8), pages 693-720.
    10. Junshan Shen & Shuyuan He, 2008. "Empirical likelihood confidence intervals for hazard and density functions under right censorship," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 575-589, September.
    11. Wang, Qi-Hua, 1999. "Some bounds for the error of an estimator of the hazard function with censored data," Statistics & Probability Letters, Elsevier, vol. 44(4), pages 319-326, October.
    12. BuHamra, Sana S. & Al-Kandari, N.M.Noriah M. & Ahmed, S. E., 2004. "Inference concerning quantile for left truncated and right censored data," Computational Statistics & Data Analysis, Elsevier, vol. 46(4), pages 819-831, July.
    13. Mokkadem, Abdelkader & Pelletier, Mariane, 2021. "A compact law of the iterated logarithm for online estimator of hazard rate under random censoring," Statistics & Probability Letters, Elsevier, vol. 178(C).
    14. Kawaguchi, Kohei, 2017. "Testing rationality without restricting heterogeneity," Journal of Econometrics, Elsevier, vol. 197(1), pages 153-171.
    15. Ülkü Gürler & Jane-Ling Wang, 1993. "Nonparametric estimation of hazard functions and their derivatives under truncation model," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 45(2), pages 249-264, June.
    16. Gürler, Ülkü & Deniz Yenigün, C., 2011. "Full and conditional likelihood approaches for hazard change-point estimation with truncated and censored data," Computational Statistics & Data Analysis, Elsevier, vol. 55(10), pages 2856-2870, October.
    17. Cai, Zongwu, 1998. "Kernel Density and Hazard Rate Estimation for Censored Dependent Data," Journal of Multivariate Analysis, Elsevier, vol. 67(1), pages 23-34, October.
    18. Sun, Liuquan, 1997. "Bandwidth choice for hazard rate estimators from left truncated and right censored data," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 101-114, December.
    19. Zhang, Biao, 1998. "A note on the integrated square errors of kernel density estimators under random censorship," Stochastic Processes and their Applications, Elsevier, vol. 75(2), pages 225-234, July.
    20. Fuxia Cheng, 2017. "Asymptotic Properties of Hazard Rate Estimator in Censored Linear Regression," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(1), pages 1-12, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:58:y:1995:i:2:p:217-245. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.