Quadratic Diffusion

  • Chongfu Huang
  • Yong Shi
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 99)


This chapter proves that the Epanechnikov function (a quadratic function) is just the optimal diffusion function in theory. Some models from the kernel theory are introduced to choose the diffusion coefficients. The golden section method is employed to search for a diffusion coefficient for estimating unimodal distributions. We use computer simulation to estimate a lognormal distribution and compare quadratic diffusion with some other estimates.


Diffusion Coefficient Discrete Fourier Transform Gaussian Kernel Diffusion Estimate Diffusion Function 


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  1. 1.
    Bowmanm A.W. (1984), An alternative method of cross-validation for the smoothing of density estimates. Biometrika, Vol. 71, pp. 353–360MathSciNetCrossRefGoogle Scholar
  2. 2.
    Epanechnikov, V.A. (1969), Nonparametric estimation of a multidimensional probability density. Theor. Probab. Appl., Vol. 14, pp. 153–158CrossRefGoogle Scholar
  3. 3.
    Govindarajulu, Z. (1981), The Sequential Statistical Analysis of Hypothesis Testing, Point and Interval Estimation, and Decision Theory, American Sciences Press, Columbus, OhioGoogle Scholar
  4. 4.
    Rosenblatt, M. (1956), Remarks on some nonparametric estimates of a density function. Ann. Math. Statist., Vol. 27, pp. 832–837MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Rudemo, M. (1982), Empirical choice of histograms and kernel density estimators. Scand. J. Statist., Vol. 9, pp. 65–78MathSciNetMATHGoogle Scholar
  6. 6.
    Silverman, B.W. (1982), Kernel density estimation using the fast Fourier transform. Statistical Algorithm AS 176. Appl. Statist., Vol. 31, pp. 93–97MATHCrossRefGoogle Scholar
  7. 7.
    Silverman, B.W. (1986), Density Estimation for Statistics and Data Analysis. Chapman and Hall, LondonMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Chongfu Huang
    • 1
  • Yong Shi
    • 2
  1. 1.Institute of Resources ScienceBeijing Normal UniversityBeijingChina
  2. 2.College of Information Science and TechnologyUniversity of Nebraska at OmahaOmahaUSA

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