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Laws of the Iterated Logarithm for the Local U-Statistic Process

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Abstract

Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L p norms for these estimators.

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References

  1. Brown, B.: Martingale central limit theorems. Ann. Math. Stat. 42, 59–66 (1971)

    MATH  Google Scholar 

  2. Deheuvels, P., Mason, D.M.: Functional laws of the iterated logarithm for local empirical processes indexed by sets. Ann. Probab. 22, 1619–1661 (1994)

    MATH  MathSciNet  Google Scholar 

  3. Deheuvels, P., Mason, D.M.: General confidence bounds for nonparametric functional estimators. Stat. Inference Stoch. Process. 7, 225–277 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Deheuvels, P., Einmahl, U., Mason, D.M.: Asymptotic independence of the local empirical process indexed by functions. In: High Dimensional Probability, II, Seattle, WA, 1999. Progr. Probab., vol. 47, pp. 183–206. Birkhäuser, Boston (2000)

    Google Scholar 

  5. de la Peña, V.H., Giné, E.: Decoupling. From Dependence to Independence. Randomly Stopped Processes. U-Statistics and Processes. Martingales and Beyond. Probability and its Applications. Springer, New York (1999)

    Google Scholar 

  6. Devroye, L., Györfi, L.: Nonparametric Density Estimation: The L1 View. Wiley, New York (1985)

    MATH  Google Scholar 

  7. Dudley, R.M.: Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics, vol. 63. Cambridge University Press, New York (1999)

    MATH  Google Scholar 

  8. Dunford, N., Schwartz, J.T.: Linear Operators, Part I, 3rd printing. Wiley-Interscience, New York (1966)

    Google Scholar 

  9. Einmahl, U., Mason, D.M.: Gaussian approximation of local empirical processes indexed by functions. Probab. Theory Relat. Fields 107, 283–311 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Einmahl, U., Mason, D.M.: Strong approximations for local empirical processes. In: High Dimensional Probability, II, Oberwolfach, 1996. Progr. Probab., vol. 43, pp. 75–02. Birkhäuser, Basel (1998)

    Google Scholar 

  11. Einmahl, U., Mason, D.M.: An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13, 1–37 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Einmahl, U., Mason, D.M.: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat. 33, 1380–1403 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  13. Frees, W.: Estimating densities of functions of observations. J. Am. Stat. Assoc. 89, 517–525 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  14. Giné, E., Guillou, A.: Rates of strong consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré 38, 907–921 (2002)

    Article  MATH  Google Scholar 

  15. Giné, E., Mason, D.M.: The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator. Bernoulli 10, 721–752 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Giné, E., Mason, D.M.: On local U-statistic processes and the estimation of densities of functions of several sample variables. Ann. Stat. 35(3) (2007, to appear)

  17. Giné, E., Latała, R., Zinn, J.: Exponential and moment inequalities for U-statistics. In: High Dimensional Probability, II, Seattle, WA, 1999. Progr. Probab., vol. 47, pp. 13–38. Birkhäuser, Boston (2000)

    Google Scholar 

  18. Giné, E., Mason, D.M., Zaitsev, A.Y.: The L 1-norm density estimator process. Ann. Probab. 31, 719–768 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kuelbs, J.: A strong convergence theorem for Banach space valued random variables. Ann. Probab. 4, 744–771 (1976)

    MATH  MathSciNet  Google Scholar 

  20. Ledoux, M., Talagrand, M.: Characterization of the law of the iterated logarithm in Banach spaces. Ann. Probab. 16, 1242–1264 (1988)

    MATH  MathSciNet  Google Scholar 

  21. Ledoux, M., Talagrand, M.: Comparison theorems, random geometry and some limit theorems for empirical processes. Ann. Probab. 17, 596–631 (1989)

    MATH  MathSciNet  Google Scholar 

  22. Ledoux, M., Talagrand, M.: Probability on Banach Spaces. Springer, Berlin (1991)

    Google Scholar 

  23. Major, P.: An estimate on the supremum of a nice class of stochastic integrals and U-statistics. Probab. Theory Relat. Fields 134, 489–537 (2006)

    MATH  MathSciNet  Google Scholar 

  24. Mason, D.M.: A uniform functional law of the logarithm for the local empirical process. Ann. Probab. 32, 1391–1418 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rost, D.: Limit theorems for smoothed empirical processes. In: High Dimensional Probability, II, Seattle, WA, 1999. Progr. Probab., vol. 47, pp. 107–113. Birkhäuser, Boston (2000)

    Google Scholar 

  26. Schick, A., Wefelmeyer, W.: Root n consistent density estimators for sums of independent random variables. J. Nonparametr. Stat. 16, 925–935 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shorack, G.R., Wellner, J.A.: Empirical Processes with Applications to Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1986)

    Google Scholar 

  28. van der Vaart, A.: Weak convergence of smoothed empirical processes. Scand. J. Stat. 21, 501–504 (1994)

    MATH  Google Scholar 

  29. van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes with Applications to Statistics. Springer, New York (1996)

    MATH  Google Scholar 

  30. Yukich, J.E.: Weak convergence of smoothed empirical processes. Scand. J. Stat. 19, 271–279 (1992)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT, 06269-3009, USA

    Evarist Giné

  2. Food and Resource Economics, University of Delaware, 206 Townsend Hall, Newark, DE, 19717, USA

    David M. Mason

Authors
  1. Evarist Giné
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  2. David M. Mason
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Correspondence to Evarist Giné.

Additional information

Research of E. Giné partially supported by NSA Grant H98230-04-1-0075.

Research of D.M. Mason partially supported by NSA Grant MDA904-02-1-0034 and NSF Grant DMS-0503908.

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Giné, E., Mason, D.M. Laws of the Iterated Logarithm for the Local U-Statistic Process. J Theor Probab 20, 457–485 (2007). https://doi.org/10.1007/s10959-007-0067-0

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  • DOI: https://doi.org/10.1007/s10959-007-0067-0

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