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Advances in Nonparametric Function Estimation

  • Theo Gasser
Part of the Lecture Notes in Statistics book series (LNS, volume 109)

Abstract

This paper has three aims: First, to demonstrate the need for nonparametric curve fitting techniques and to provide an introduction to such statistical techniques. Second, whatever method we want to apply, selection of a smoothing parameter or bandwidth is required. It would be desirable to determine the appropriate bandwidth from the data themselves, tuned to the problem at hand. Considerable progress has been made in recent years, most notably by the introduction of “plug-in” selectors, and the essence of this development is summarized. Third, a variety of function estimators has been discussed in the literature. Quite recently, local polynomial fitting stirred a lot of interest, seemed to become the ultimate answer. The potential pitfalls of these estimators and some remedies are discussed.

Key words and phrases

Curve estimation nonparametric regression bandwidth choice, kernel estimators local polynomials 

AMS 1991 subject classifications

Primary 62G07 secondary 62G20 

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References

  1. [1]
    Brockmann, M., Gasser, Th. and Herrmann, E. (1993): Locally adaptive bandwidth choice for kernel regression estimators. J. Amer. Statist. Assoc. 88 1302–1309.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chu, C.K. and Marron, J.S. (1991): Choosing a kernel regression estimator. Statistical Science 6 404–433.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Devroye, L.P. (1978): The uniform convergence of nearest neighbor regression function estimators and their application in optimization. IEEE Trans. Inform. Theory IT-24 142–151.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Donoho, D., Johnstone, I., Kerkyacharian, G. and Picard, D. (1994): Wavelet Shrinkage: Asymptopia? J. Roy. Statist. Soc., to appear.Google Scholar
  5. [5]
    Fan, J. (1993): Local linear regression smoothers and their minimax efficiencies. Ann. Statist. 21 196–216.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Fan, J. and Gijbels, I. (1992): Variable bandwidth and local linear regression smoothers. Ann. Statist. 20 2008–2036.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Gasser, Th. and Müller, H.-G. (1979): Kernel estimation of regression functions. Smoothing techniques for curve estimation. Lecture Notes in Mathematics #757 23–68. New York, Springer.CrossRefGoogle Scholar
  8. [8]
    Gasser, Th. and Müller, H.-G. (1984): Estimating regression functions and their derivatives by the kernel method. Scand. J. Statist. 11 171–185.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Gasser, Th., Müller, H.-G. and Mammitzsch, V. (1985): Kernels for non-parametric curve estimation. J. Roy. Statist. Soc. B 47 238–252.zbMATHGoogle Scholar
  10. [10]
    Gasser, Th., Sroka, L. and Jennen-Steinmetz, Ch. (1986): Residual variance and residual pattern in nonlinear regression. Biometrika 73 625–633.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Gasser, Th., Kneip, A., and Köhler, W. (1991): A flexible and fast method for automatic smoothing. J. Amer. Statist. Assoc. 86 643–652.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Granowski, B.L. and Müller, H.-G. (1989): On the optimality of a class of polynomial kernels. Statist. and Decisions 7 301–312.MathSciNetGoogle Scholar
  13. [13]
    Hall, P., Sheather, S.J., Jones, M.C. and Marron, J.S. (1991): On optimal data-based bandwidth selection in kernel density estimation. Biometrika 78 263–269.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Hall, P. and Marron, J.S. (1991): Lower bounds for bandwidth selection in density estimation. Probab. Th. Rel. Fields 90 149–173.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Hardie, W., Hall, P. and Marron, J.S. (1988): How far are automatically chosen regression smoothing parameters from their optimum? (with discussion). J. Amer. Statist. Assoc. 83 86–101.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Härdie, W. (1990): Applied Nonparametric Regression. Cambridge Univ. Press, Cambridge.Google Scholar
  17. [17]
    Hart, J.D. (1991): Kernel regression estimation with time series errors. J. Roy. Statist. Soc. B 53 173–188.zbMATHGoogle Scholar
  18. [18]
    Herrmann, E., Gasser, Th. and Kneip, A. (1992): Choice of bandwidth for kernel regression when residuals are correlated. Biometrika 79 783–796.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    Jennen-Steinmetz, Ch. and Gasser, Th. (1988): A unifying approach to nonparametric regression estimation. J. Amer. Statist. Assoc. 83 1084–1089.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Macauley, F.R. (1931): The Smoothing of Time Series. National Bureau of Economic Research, New York.Google Scholar
  21. [21]
    Müller, H.-G. and Stadtmüller, U. (1987): Variable bandwidth kernel estimators of regression curves. Ann. Statist. 15 182–201.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    Nadaraya, E.A. (1964): On estimating regression. Theory Probab. Appl. 9 141–142.CrossRefGoogle Scholar
  23. [23]
    Reinsch, C.H. (1967): Smoothing by spline functions. Numer. Math. 10 177–183.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Seifert, B. and Gasser, Th. (1994): Finite Sample Variance of Local Polynomials: Analysis and Solutions. Manuscript.Google Scholar
  25. [25]
    Seifert, B., Brockmann, M., Engel, J. and Gasser, T. (1994): Fast algorithms for nonparametric curve estimation. J. Comp. Graph. Statist. 3 192–213.MathSciNetCrossRefGoogle Scholar
  26. [26]
    Sheather, S. and Jones, M.C. (1991): A reliable data-based bandwidth selection method for kernel density estimation. J. Roy. Statist. Soc. B 53 683–690.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Wang, K. and Gasser, Th. (1994): Optimal Rates of Convergence for Local Bandwidth Selection. Manuscript.Google Scholar
  28. [28]
    Watson, G.S. (1964): Smooth regression analysis. Sankhyā A 26 359–372.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1996

Authors and Affiliations

  • Theo Gasser
    • 1
  1. 1.University of ZürichSwitzerland

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