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Estimating Distributions with a Fixed Number of Modes

  • Martin B. Mächler
Part of the Lecture Notes in Statistics book series (LNS, volume 109)

Abstract

A new approach for non- or semi-parametric density estimation allows to specify modes and antimodes. The new smoother is a Maximum Penalized Likelihood (MPL) estimate with a novel roughness penalty. It penalizes a relative change of curvature which allows considering modes and inflection points. For a given number of modes, the score function, \(l^\prime = (\log f)^\prime\) can be represented as \({l^\prime }\left( x \right) = \pm (x - {w_1}) \cdots (x - {w_m})\cdot{\text{ }}exp{h_l}(x)\), a semiparametric term with parameters w j (model order m) and nonparametric part h l(·). The MPL variational problem is equivalent to a differential equation with boundary conditions. The exponential and normal distributions are smoothest limits of the new estimator for zero and one mode, respectively.

Key words and phrases

Nonparametric semi-parametric density estimation smoothing roughness penalty maximum penalized likelihood inflection point boundary value problem differential equation unimodality multimodality 

AMS 1991 subject classifications

Primary 62G07 secondary 34B10 41A29 65D07 65D10 

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References

  1. [1]
    Ascher, U.M., Mattheij, R. M.M. and Russell, R.D. (1988): Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. Prentice-Hall series in computational mathematics. Prentice Hall, Englewood Cliffs, New Jersey.Google Scholar
  2. [2]
    Bader, G. and Ascher, U. (1987): A new basis implementation for a mixed order boundary value ODE solver. SIAM Journal on Scientific and Statistical Computing 8 483–500.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Cox, D.R. (1966): Notes on the analysis of mixed frequency distributions. J. Math. Statist. Psychol. 19 39–47.CrossRefGoogle Scholar
  4. [4]
    Good, I.J. and Gaskins, R.A. (1971): Non-parametric roughness penalties for probability densities. Biometrika 58 255–277.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Good, I.J. and Gaskins, R.A. (1980): Density estimation and bump-hunting by the penalized likelihood method exemplified by scattering and meteorite data. J. Amer. Statist. Assoc. 75 42–73. Comments & Rejoinder.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Härdie, W. (1990): Smoothing Techniques With Implementation in S. Springer-Verlag, Berlin.Google Scholar
  7. [7]
    Hartigan, J.A. and Hartigan, P.M. (1985): The dip test of unimodality. Ann. Statist. 13 70–84.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Izenman, A.J. (1991): Recent developments in nonparametric density estimation. J. Amer. Statist. Assoc. 86(413) 205–224.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Izenman, A.J. and Sommer, C.J. (1988). Philatelic mixtures and multimodal densities. J. Amer. Statist. Assoc. 83 941–953.CrossRefGoogle Scholar
  10. [10]
    Kooperberg, C. and Stone, C.J. (1992): Logspline density estimation for censored data. Journal of Computational and Graphical Statistics 1(4) 301–328.CrossRefGoogle Scholar
  11. [11]
    Mächler, M. (1989): ‘Parametric’ Smoothing Quality in Nonparametric Regression: Shape Control by Penalizing Inflection Points. Ph.D. Thesis, no 8920, ETH Zürich, Switzerland.Google Scholar
  12. [12]
    Mächler, M. (1995): Very smooth nonparametric curve estimation by penalizing change of curvature. Ann. Statist 23(5), to appear.Google Scholar
  13. [13]
    Minnotte, M.C. and Scott, D.W. (1993): The mode tree: A tool for visualization of nonparametric density features. J. of Computational and Graphical Statistics 2(1) 51–68.CrossRefGoogle Scholar
  14. [14]
    Silverman, B.W. (1981): Using kernel density estimates to investigate multimodality. J. Roy Statist. Soc. B 43 97–99.Google Scholar
  15. [15]
    Silverman, B.W. (1986): Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.zbMATHGoogle Scholar
  16. [16]
    Tapia, R.A. and Thompson, J.R. (1978): Nonparametric Probability Density Estimation. John Hopkins Univ. Press, Baltimore.zbMATHGoogle Scholar
  17. [17]
    Thompson, J.R. and Tapia, R.A. (1990): Nonparametric Function Estimation, Modeling, and Simulation. SIAM, Philadelphia.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Martin B. Mächler
    • 1
  1. 1.ETH ZürichSwitzerland

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