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Journal of Theoretical Probability

, Volume 26, Issue 3, pp 697–721 | Cite as

Uniform-in-Bandwidth Functional Limit Laws

  • Paul Deheuvels
  • Sarah Ouadah
Article

Abstract

We provide uniform-in-bandwidth functional limit laws for the increments of the empirical and quantile processes. Our theorems, established in the framework of convergence in probability, imply new sharp uniform-in-bandwidth limit laws for functional estimators. In particular, they yield the explicit value of the asymptotic limiting constant for the uniform-in-bandwidth sup-norm of the random error of kernel density estimators. We allow the bandwidth to vary within the complete range for which the estimators are consistent.

Keywords

Functional limit laws Kernel density estimators Nonparametric functional estimators Convergence in probability Weak laws Laws of large numbers 

Mathematics Subject Classification (2000)

62G05 62G07 62G20 62G30 60F15 60F17 

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References

  1. 1.
    Berkes, I., Philipp, W.: Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7, 29–54 (1979) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Blondin, D.: Estimation nonparamétrique multidimensionnelle des dérivées de la régression. C. R. Acad. Sci. Paris, Math. 339, 713–716 (2004) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Blondin, D.: Lois limites uniformes et estimation non-paramétrique de la régression. Doctoral Dissertation, Université Pierre et Marie Curie, Dec. 10, 2004, Paris, France (2004) Google Scholar
  4. 4.
    Deheuvels, P., Mason, D.M.: Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20, 1248–1287 (1992) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Deheuvels, P.: Functional laws of the iterated logarithm for large increments of empirical and quantile processes. Stoch. Process. Appl. 43, 133–163 (1992) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Deheuvels, P.: Strong laws for local quantile processes. Ann. Probab. 25, 2007–2054 (1997) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Deheuvels, P., Einmahl, J.H.J.: Functional limit laws for the increments of Kaplan–Meier product-limit processes and applications. Ann. Probab. 28, 1301–1335 (2000) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Deheuvels, P., Mason, D.M.: General asymptotic confidence bands based on kernel-type function estimators. Stat. Inference Stoch. Process. 7, 225–277 (2004) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dony, J.: Nonparametric regression estimation—An empirical process approach to uniform in bandwidth consistency of kernel-type estimators and conditional U-statistics. Doctoral Dissertation, Vrije Universiteit Brussel, Belgium (2008) Google Scholar
  10. 10.
    Dony, J., Einmahl, U.: Weighted uniform consistency of kernel density estimators with general bandwidth sequences. Electron. J. Probab. 11, 844–859 (2006) MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dony, J., Einmahl, U.: Uniform in bandwidth consistency of kernel-type estimators at a fixed point. IMS Collections 5, 308–325 (2009) MathSciNetGoogle Scholar
  12. 12.
    Dony, J., Einmahl, U., Mason, D.M.: Uniform in bandwidth consistency of local polynomial regression function estimators. Aust. J. Stat. 35, 105–120 (2006) Google Scholar
  13. 13.
    Dony, J., Mason, D.M.: Uniform in bandwidth consistency of conditional U-statistics. Bernoulli 4, 1108–1133 (2008) MathSciNetCrossRefGoogle Scholar
  14. 14.
    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) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Einmahl, U., Mason, D.M.: Uniform in bandwidth consistency of kernel-type function estimators. Ann. Stat. 33, 1380–1403 (2005) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent rv’s and the sample df. I. Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975) MATHCrossRefGoogle Scholar
  17. 17.
    Mason, D.M.: A strong limit theorem for the oscillation modulus of the uniform empirical process. Stoch. Process. Appl. 17, 127–136 (1984) MATHCrossRefGoogle Scholar
  18. 18.
    Mason, D.M.: A uniform functional law of the logarithm for the local empirical process. Ann. Probab. 32, 1391–1418 (2004) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mason, D.M.: Proving consistency of non-standard kernel estimators. Statist. Infer. Stoch. Processes. (2011, to appear) Google Scholar
  20. 20.
    Mason, D.M., Swanepoel, J.: A general result on the uniform in bandwidth consistency of kernel-type function estimators. Test 20, 72–94 (2011) MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mason, D.M., Shorack, G.R., Wellner, J.A.: Strong limit theorems for oscillation moduli of the empirical process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 65, 93–97 (1983) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Parzen, E.: On the estimation of a probability density function and mode. Ann. Math. Stat. 33, 1065–1076 (1962) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat. 27, 832–837 (1956) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Schilder, M.: Asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc. 125, 63–85 (1966) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Shorack, G.R.: Kiefer’s theorem via the Hungarian construction. Z. Wahrscheinlichkeitstheor. Verw. Geb. 61, 369–373 (1982) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Silverman, B.: Weak and strong consistency of the kernel estimate of a density and its derivatives. Ann. Stat. 6, 177–184 (1978). (Addendum 8, 1175–1176 (1980)) MATHCrossRefGoogle Scholar
  27. 27.
    Stute, W.: The oscillation behavior of empirical processes. Ann. Probab. 10, 86–107 (1982) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Stute, W.: A law of the iterated logarithm for kernel density estimators. Ann. Probab. 10, 414–422 (1982) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Varron, D.: Lois fonctionnelles uniforme du logarithme itéré pour les accroissements du processus empirique généralisé. Lois limites de type Chung-Mogulskii pour le processus empirique uniforme local. Doctoral Dissertation, Université Pierre et Marie Curie, Dec. 17, 2004, Paris, France (2004) Google Scholar
  30. 30.
    Varron, D.: A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process. Electron. J. Stat. 2, 1043–1064 (2008) MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Varron, D., van Keilegom, I.: Uniform in bandwidth exact rates for a class of kernel estimators. Ann. Inst. Stat. Math. (2010). doi: 10.1007/s10463-010-0286-5 Google Scholar
  32. 32.
    Viallon, V.: Functional limit laws for the increments of the quantile process with applications. Electron. J. Stat. 1, 496–518 (2007) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.L.S.T.A.Université Pierre et Marie Curie (Paris 6)Bourg-la-ReineFrance
  2. 2.L.S.T.A.Université Pierre et Marie Curie (Paris 6)Paris Cedex 05France

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