Uniform Limit Laws for Kernel Density Estimators on Possibly Unbounded Intervals

  • Paul Deheuvels
Part of the Statistics for Industry and Technology book series (SIT)


We establish uniform limit laws for kernel density estimators with minimal assumptions upon the kernel and the density.

Keywords and phrases

Density estimation nonparametric estimators empirical processes strong limit theorems weak laws of large numbers empirical processes functional laws of the iterated logarithm 


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  1. 1.
    Akaike, H. (1954). An approximation to the density function. Ann. Inst. Statist Math. 6 127–132.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bertrand-Retali, M. (1978). Convergence uniforme d’un estimateur de la densité par la méthode du noyau. Rev. Roumaine de Math. Pures Appl. 23 361–385MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bosq, D. and Lecoutre, J. P. (1987). Théorie de l’Estimation Fonctionnelle. Economica, ParisGoogle Scholar
  5. 5.
    Chen, K. and Lo, S.-H. (1997). On the rate of uniform convergence of the product-limit estimator: strong and weak laws. Ann. Statist. 25 1050–1087MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    CsörgŐ, M. and Révész, P. (1979). How big are the increments of a Wiener process? Ann. Probab. 7 731–737MathSciNetCrossRefGoogle Scholar
  7. 7.
    CsörgŐ, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New YorkGoogle Scholar
  8. 8.
    Deheuvels, P. (1974). Conditions nécessaires et suffisantes de convergence presque sûre et uniforme presque sûre des estimateurs de la densité. C.R. Acad. Sci. Paris. Ser.A 278 1217–1220MathSciNetzbMATHGoogle Scholar
  9. 9.
    Deheuvels, P. (1992). Functional laws of the iterated logarithm for large increments of empirical and quantile processes. Stochastic Process. Appl. 43 133–163MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Deheuvels, P. (1999). Limit laws for kernel density estimators for kernels with unbounded supports. Probability Theory. and Mathematical Statistics. M. L. Puri ed. (to appear).Google Scholar
  11. 11.
    Deheuvels, P. and Einmahl, J. H. J. (1996). On the strong limiting behavior of local functionals of empirical processes based upon censored data. nn. Probab. 24 504–525.MathSciNetzbMATHGoogle Scholar
  12. 12.
    Deheuvels, P. and Einmahl, J. H. J. (1999). Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications. Ann. Probab. (to appear).Google Scholar
  13. 13.
    Deheuvels, P. and Mason, D. M. (1992). Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab. 20 1248–1287MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Devroye, L. (1987). A Course in Density Estimation. Birkhäuser, BostonzbMATHGoogle Scholar
  15. 15.
    Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The L 1 View. Wiley, New YorkzbMATHGoogle Scholar
  16. 16.
    Einmahl, U. and Mason, D. M. (1997). Gaussian approximation of local empirical processes indexed by functions. Probab. Theor. Rel. Fields. 107 283–301MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Einmahl, U. and Mason, D. M. (1998). Strong approximation to the local empirical process. Progress in Probability. 43 75–92. Birkhäuser Verlag, BaselMathSciNetGoogle Scholar
  18. 18.
    Hall, P. (1981). Laws of the iterated logarithm for nonparametric density estimators. Z. Wahrscheinlichkeit. Verw. Gebiete 56 47–61zbMATHCrossRefGoogle Scholar
  19. 19.
    Hall, P. (1990). On the law of the iterated logarithm for density estimators. Stat. Probab. Lett. 9 237–240zbMATHCrossRefGoogle Scholar
  20. 20.
    Konakov, V. (1972). Nonparametric estimation of density functions. Theor. Probab. Appl. 17 361-362Google Scholar
  21. 21.
    Konakov, V. and Piterbarg, V. (1984). Rate of convergence of distributions of maximal deviations of Gaussian processes and empirical density functions. II. Theor. Probab. Appl. 28 172–178zbMATHCrossRefGoogle Scholar
  22. 22.
    Kuelbs, J. (1976). Estimation of the multidimensional probability density function. MRC Technical Report. 1646. University of Wisconsin, MadisonGoogle Scholar
  23. 23.
    Mason, D. M. (1984). A strong limit theorem for the oscillation modulus of the uniform empirical quantile process. Stoch. Processes Appl. 17 127–136zbMATHCrossRefGoogle Scholar
  24. 24.
    Mason, D. M., Shorack, G. R. and Wellner, J. A. (1983). Strong limit theorems for oscillation moduli of the uniform empirical process. Z. Wahrscheinlichkeit. Verw. Gebiete 65 83–97MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Parzen, E. (1962). On the estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Peligrad, M. (1992). Properties of uniform consistency of the kernel estimators of density and of regression functions under dependence assumptions. Stochastics and Stochastics Reports. 40 147–168MathSciNetzbMATHGoogle Scholar
  27. 27.
    Reiss, R. D. (1975). Consistency of a certain class of empirical density functions. Metrika 22 189–203MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Révész, P. (1978). A strong law of the empirical density function. Peri-odica Math. Hungar. 9 317–324zbMATHCrossRefGoogle Scholar
  29. 29.
    Révész, P. (1982). On the increments of Wiener and related processes. Ann. Probab. 10 613–622.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832–837MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Roussas, G. (1990). Nonparametric Functional Estimation and Related Topics. NATO ASI Series 335. Kluwer, DordrechtGoogle Scholar
  32. 32.
    Scott, D. W. (1992). Multivariate Density Estimation — Theory, Practice and Visualization. Wiley, New YorkzbMATHCrossRefGoogle Scholar
  33. 33.
    Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New YorkzbMATHGoogle Scholar
  34. 34.
    Silverman, B. W. (1978). Weak and strong uniform consistency of the kernel estimate of a density function and its derivatives. Ann. Statist. 6 177–184 [Addendum (1980). Ann. Statist. 8 1175-1776CrossRefGoogle Scholar
  35. 35.
    Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman and Hall, LondonzbMATHGoogle Scholar
  36. 36.
    Singh, R. S. (1977). Improvements on some known nonparametric uniformly consistent estimators of derivatives of a density. Ann. Statist. 5 394–399MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Strassen, V. (1964). An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeit. Verw. Gebiete 3 211–226MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Stute, W. (1982a). The oscillation behavior of empirical processes. Ann. Probab. 10 86–107MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Stute W. (1982b). A law of the iterated logarithm for kernel density estimators. Ann. Probab. 10 414–422Google Scholar
  40. 40.
    Stute, W. (1984). The oscillation behavior of empirical processes: the multivariate case. Ann. Probab. 12 361–379MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Tanner, M. and Wong, W. (1983). The estimation of the hazard function from randomly censored data by the kernel method. Ann. Statist. 11 422–439MathSciNetGoogle Scholar
  42. 42.
    Watson, G. S. and Leadbetter, M. R. (1964a). Hazard analysis. J. Biometrika 51 175–184MathSciNetzbMATHGoogle Scholar
  43. 43.
    Watson, G. S. and Leadbetter, M. R. (1964b). Hazard analysis. II. Sankhyā Ser. A 26 101–116MathSciNetzbMATHGoogle Scholar
  44. 44.
    Woodroofe, M. (1967). On the maximum deviation of the sample density. Ann. Math. Statist. 38 475–481MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2000

Authors and Affiliations

  • Paul Deheuvels
    • 1
  1. 1.Université Paris VIFrance

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