Computational Statistics

, Volume 16, Issue 2, pp 271–298 | Cite as

Numerical results concerning a sharp adaptive density estimator

  • Cristina ButuceaEmail author


We present here a simulation study of the behavior of a particular kernel density estimator. It was previously proven that this nonparametric estimator is sharp in the sense of the minimax adaptive theory, which means that it is equally well performing for very smooth or unsmooth densities. The method selects locally both the bandwidth and the kernel function according to the evaluated smoothness of the underlying density. In this paper we describe the method and apply it successfully to i.i.d. simulated data of different probability densities.


Pointwise density estimation adaptivity kernel estimator Lepski’s criterion simulation study 


  1. [1]
    Abramson, I.S. (1982) On bandwidth variation in kernel estimates-a square root law. Ann. Statist. 10 1217–1223MathSciNetCrossRefGoogle Scholar
  2. [2]
    Berlinet, A. and Devroye, L. (1994) A comparison of kernel density estimates. Puhl. Inst. Statist. Univ. Paris 38 3–59MathSciNetzbMATHGoogle Scholar
  3. [3]
    Breiman, L., Meisel, W. and Purcell, E. (1977) Variable kernel estimates of multivariate densities. Technometrics 19 135–144CrossRefGoogle Scholar
  4. [4]
    Butucea, C. (1999a) Nonparametric adaptive estimation of a probability density; rates of convergence, exact constant and numerical results PhD Thesis, Paris 6 UniversityGoogle Scholar
  5. [5]
    Butucea, C. (1999b) Constante exacte adaptative dans l’estimation de la densité. Comptes Rendus Acad. Sei., Série I 329 535–540MathSciNetzbMATHGoogle Scholar
  6. [6]
    Chaudhuri, P. and Marron, J.S. (1997) SiZer for exploration of structures in curves. North Carolina Inst, of Statist. Mimeo Series 2355Google Scholar
  7. [7]
    Devroye, L. and Gyorfi, L. (1985) Nonparametric Density Estimation: The \(\mathbb{L}_{1}\) View. J. Wiley, New YorkzbMATHGoogle Scholar
  8. [8]
    Devroye, L. and Lugosi, G. (1996) A universally acceptable smoothing factor for kernel density estimates. Ann. Statist. 24 2499–2512MathSciNetCrossRefGoogle Scholar
  9. [9]
    Devroye, L. and Lugosi, G. (1997) Nonasymptotic universal smoothing factors, kernel complexity and Yatracos classes. Ann. Statist. 25 2626–2637MathSciNetCrossRefGoogle Scholar
  10. [10]
    Devroye, L. and Lugosi, G. (1998) Variable kernel estimates: on the impossibilty of tunning the parameters. ManuscriptGoogle Scholar
  11. [11]
    Devroye, L., Lugosi, G. and Udina, F. (1998) Inequalities for anew data-based method for selecting nonparametric density estimates. ManuscriptGoogle Scholar
  12. [12]
    Farmen, M. and Marron, J.S. (1999) An assessment of finite sample performance of adaptive methods in density estimation. Comp. Statist. Data Anal. 30 143–168CrossRefGoogle Scholar
  13. [13]
    Fryer, M. J. (1976) Some errors associated with the non-parametric estimation of density functions. J. Inst. Maths Applies 18 371–380CrossRefGoogle Scholar
  14. [14]
    Gradshteyn, I.S. and Ryzhik, I.M. (1994) Table of Integrals, Series and Products. Academic Press 5th EditionGoogle Scholar
  15. [15]
    Hall, P., Hu, T.C. and Marron, J.S. (1995) Improved variable window kernel estimates of probabilty densities. Ann. Statist. 23 1–10MathSciNetCrossRefGoogle Scholar
  16. [16]
    Hall, P. and Marron, J.S. (1988) Variable window width kernel estimates of probabilty densities. Probab. Th. Rel. Fields 80 37–49CrossRefGoogle Scholar
  17. [17]
    Hall, P., Marron, J.S. and Park, B.U. (1992) Smoothed cross-validation. Probab. Theory Rel. Fields 92 1–20MathSciNetCrossRefGoogle Scholar
  18. [18]
    Hall, P. and Schucany W.R. (1989) A local cross-validation algorithm. Statist. Probab. Letters 8 107–117MathSciNetzbMATHGoogle Scholar
  19. [19]
    Hazelton, M. (1996) Bandwidth selection for local density estimators. Scandin. Journal of Statist. 23 221–232MathSciNetzbMATHGoogle Scholar
  20. [20]
    Jones, M.C., Marron, J.S. and Sheather, S.J. (1996) Progress in data-based bandwidth selection for kernel density estimation. Comp. Statist. 11: 337–381MathSciNetzbMATHGoogle Scholar
  21. [21]
    Lepskii, O.V. (1990) On a problem of adaptive estimation in Gaussian white noise. Theory Prob. Appl. 35 454–466MathSciNetCrossRefGoogle Scholar
  22. [22]
    Marron, J.S. and Tsybakov, A.B. (1995) Visual Error Criteria for Qualitative Smoothing. J. Amer. Statist. Assoc. 90 499–507MathSciNetCrossRefGoogle Scholar
  23. [23]
    Marron, J.S. and Wand M.P. (1992) Exact mean integrated square error. Ann. Statist. 20 712–713MathSciNetCrossRefGoogle Scholar
  24. [24]
    Mielniczuk, J., Sarda, P. and Vieu Ph. (1989) Local data-driven bandwidth choice for density estimation. J. Statist. Planning Infer. 23 53–69MathSciNetCrossRefGoogle Scholar
  25. [25]
    Parzen, E. (1962) On the estimation of probability density function and mode. Ann. Math. Statist. 33 1065–1076MathSciNetCrossRefGoogle Scholar
  26. [26]
    Rosenblatt, M. (1956) On some nonparametric estimates of a density function. Ann. Math. Statist. 27 832–837MathSciNetCrossRefGoogle Scholar
  27. [27]
    Rudemo, M. (1982) Empirical choice of histograms and kernel density estimators. Scandin. J. Statist. 9 65–78MathSciNetzbMATHGoogle Scholar
  28. [28]
    Sain, S.R., Baggerly, K.A. and Scott, D.W. (1994) Cross-validation of multivariate densities. Ann. Statist. 89 807–817MathSciNetzbMATHGoogle Scholar
  29. [29]
    Sain, S.R. and Scott, D.W. (1996) On locally adaptive density estimation. J. Amer. Statist. Assoc. 436 1525–1534MathSciNetCrossRefGoogle Scholar
  30. [30]
    Scott, D.W. (1992) Multivariate density estimation: theory, practice and visualization. J. Wiley, New YorkCrossRefGoogle Scholar
  31. [31]
    Sheather, S. (1986) An improved data-based algorithm for choosing the window width when estimating the density at a point. Comp. Statist. Data Anal. 4 61–65CrossRefGoogle Scholar
  32. [32]
    Sheather, S.J. and Jones M.C. (1991) A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc., Ser B 53 683–690MathSciNetzbMATHGoogle Scholar
  33. [33]
    Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman and Hall, New YorkCrossRefGoogle Scholar
  34. [34]
    Stone, C.J. (1984) An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285–1297MathSciNetCrossRefGoogle Scholar
  35. [35]
    Terrell, G.R. and Scott, D.W. (1992) Variable kernel density estimation. Ann. Statist. 20 1236–1265MathSciNetCrossRefGoogle Scholar
  36. [36]
    Tsybakov, A.B. (1998) Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. Ann. Statist. 26Google Scholar
  37. [37]
    Wand, M.P. and Jones, M.C. (1995) Kernel smoothing. Chapman and Hall, London Computational Statistics (2001) 16:299–312zbMATHGoogle Scholar

Copyright information

© Physica-Verlag 2001

Authors and Affiliations

  1. 1.Humboldt Universität zu BerlinBerlinGermany
  2. 2.Paris 6 UniversityParisFrance

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