Measurement Techniques

, Volume 60, Issue 6, pp 515–522 | Cite as

Analysis of Optimization Methods for Nonparametric Estimation of the Probability Density with Respect to the Blur Factor of Kernel Functions

GENERAL PROBLEMS OF METROLOGY AND MEASUREMENT TECHNIQUE
First Online:
Received:
  • 38 Downloads

The results of a comparison of the most common optimization methods for the nonparametric estimation of the probability density of Rosenblatt–Parzen are presented. To select the optimal values of the blur coefficients of kernel functions, minimum conditions for the standard deviation of the nonparametric estimate of the probability density and the maximum of the likelihood function are used.

Keywords

probability density nonparametric Rosenblatt–Parzen estimate optimization methods kernel function approximation properties 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    E. Parzen, “On the estimation of a probability density function and mode,” Ann. Math. Stat., 33, No. 3, 1065–1076 (1962).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    V. A. Epanechnikov, “Nonparametric estimation of a multi-dimensional probability density,” Teor. Ver. Primen., 14, No. 1, 156–161 (1969).MathSciNetMATHGoogle Scholar
  3. 3.
    A. V. Lapko, V. A. Lapko, and M. A. Sharkov, “Nonparametric methods for detecting regularities in conditions of small samples,” Izv. Vuz. Priborostr., 51, No. 8, 62–67 (2008).Google Scholar
  4. 4.
    A. V. Lapko and V. A. Lapko, “Regression estimate of the multidimensional probability density and its properties,” Opt., Inst. Data Proc., 50, No. 2, 148–153 (2014).Google Scholar
  5. 5.
    A. V. Lapko, A. V. Medvedev, and E. A. Tishina, “Optimization of nonparametric estimations,” Algorithms and Programs for Automation Systems for Experimental Research: Coll. Sci. Works, Frunze (1975), pp. 105–116.Google Scholar
  6. 6.
    M. Rudemo, “Empirical choice of histogram and kernel density estimators,” Scand. J. Stat., No. 9, 65–78 (1982).Google Scholar
  7. 7.
    P. Hall, “Large-sample optimality of least squares cross-validation in density estimation,” Ann. Stat., 11, 1156–1174 (1983).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. Jiang and S. B. Provost, “A hybrid bandwidth selection methodology for kernel density estimation,” J. Stat. Comp. Simul., 84, No. 3, 614–627 (2014).MathSciNetCrossRefGoogle Scholar
  9. 9.
    S. Dutta, “Cross-validation revisited,” Communic. Stat. Simul. Comput., 45, No. 2, 472–490 (2016).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    N. B. Heidenreich, A. Schindler, and S. Sperlich, “Bandwidth selection for kernel density estimation: a review of fully automatic selectors,” AStA Adv. Stat. Anal., 97, No. 4, 403–433 (2013).MathSciNetCrossRefGoogle Scholar
  11. 11.
    Qi Li and J. S. Racine, Nonparametric Econometrics: Theory and Practice, Princeton Univ., Princeton (2007).MATHGoogle Scholar
  12. 12.
    R. P. W. Duin, “On the choice of smoothing parameters for Parzen estimators of probability density functions,” IEE Trans. Comp., C-25, 1175–1179 (1976).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Z. I. Botev and D. P. Kroese, “Non-asymptotic bandwidth selection for density estimation of discrete data,” Method. Comp. Appl. Probab., 10, No. 3, 435–451 (2008).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    L. Devroi and L. Dierfi , Nonparametric Estimation of Density (L 1 approach) [Russian translation], Mir, Moscow (1988).Google Scholar
  15. 15.
    A. V. Lapko and V. A. Lapko, “Nonparametric pattern recognition algorithms for random values of fuzziness factors of kernel functions,” Opt. Instrum. Data Proc., 43, No. 5, 425–432 (2007).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Institute of Computational Modeling, Siberian BranchRussian Academy of SciencesKrasnoyarskRussia
  2. 2.Reshetnev Siberian State University of Science and TechnologyKrasnoyarskRussia

Personalised recommendations