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Hierarchies of higher order kernels
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  • Published: December 1993

Hierarchies of higher order kernels

  • Alain Berlinet1 

Probability Theory and Related Fields volume 94, pages 489–504 (1993)Cite this article

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Summary

Recent literature on functional estimation has shown the importance of kernels with vanishing moments although no general framework was given to build kernels of increasing order apart from some specific methods based on moment relationships. The purpose of the present paper is to develop such a framework and to show how to build higher order kernels with nice properties and to solve optimization problems about kernels. The proofs given here, unlike standard variational arguments, explain why some hierarchies of kernels do have optimality properties. Applications are given to functional estimation in a general context. In the last section special attention is paid to density estimates based on kernels of order (m, r), i.e., kernels of orderr for estimation of derivatives of orderm. Convergence theorems are easily derived from interpretation by means of projections inL 2 spaces.

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References

  • Berlinet, A.: Reproducing kernels and finite order kernels. In: Roussas, G. (ed.) Nonparametric functional estimation and related topics, pp. 3–18. London New York: Kluwer 1990

    Google Scholar 

  • Berlinet, A., Devroye, L.: Estimation d'une densité: un point sur la méthode du noyau. Stat. Anal. Données,14 (no 1), 1–32 (1989)

    Google Scholar 

  • Brézinski, C.: Padé-type approximation and general orthogonal polynomials. Basel: Birkhäuser 1980

    Google Scholar 

  • Deheuvels, P.: Estimation non-paramétrique de la densité par histogrammes généralisés. Rev. Stat. Appl.25, 5–42 (1977)

    Google Scholar 

  • Devroye, L.: The double kernel method in density estimation. Ann. Inst. Henri Poincaré25 (no 4), 553–580 (1989)

    Google Scholar 

  • Epanechnikov, V.A.: Nonparametric estimation of a multidimensional probability density. Theory Probab. Appl.14, 153–158 (1969)

    Google Scholar 

  • Freud, G.: On polynomial approximation with the weight exp (−x 2k/2). Acta Math. Acad. Sci. Hung.24, 363–371 (1973)

    Google Scholar 

  • Gasser, T., Müller, H.G.: Kernel estimation of regression function. In: Gasser, T., Rosenblatt, M. (eds.) Smoothing techniques for curve estimation. (Lect. Notes Math., vol. 757, pp. 23–68) Berlin Heidelberg New York: Springer 1979a

    Google Scholar 

  • Gasser, T., Müller, H.G.: Optimal convergence properties of kernel estimates of derivatives of a density function. In: Gasser, T., Rosenblatt, M. (eds.) Smoothing techniques for curve estimation. (Lect. Notes Math., vol. 757, pp. 144–154) Berlin Heidelberg New York: Springer 1979b

    Google Scholar 

  • Gasser, T., Müller, H.G., Mammitzsch, V.: Kernels for nonparametric curve estimation. J. R. Stat. Soc., Ser. B47, 238–252 (1985)

    Google Scholar 

  • Granovsky, B.L., Müller, H.G.: On the optimality of a class of polynomial kernel functions. Stat. Decis.7, 301–312 (1989)

    Google Scholar 

  • Granovsky, B.L., Müller, H.G.: Optimizing kernel methods: a unifying variational principle. Int. Stat. Rev.59 (3), 373–388 (1991)

    Google Scholar 

  • Hall, P., Marron, J.S.: Choice of kernel order in density estimation. Ann. Stat.16, 161–173 (1988)

    Google Scholar 

  • Jones, M.C.: Changing kernels' orders. (Preprint 1990)

  • Mc Diarmid, C.: On the method of bounded differences. In: Surveys in combinatorics. (Lond. Math. Soc. Lect. Notes Ser., vol. 141, pp. 148–188) Cambridge: Cambridge University Press 1989

    Google Scholar 

  • Mammitzsch, V.: A note on kernels of orderv, k. In: Mande, P., HuÅ¡kovich, M. (eds.) Proceedings of the Fourth Prague Symposium on Asymptotic Statistics, pp. 411–412. Prague: Charles University 1989

    Google Scholar 

  • Müller, H.G.: Smooth optimum kernel estimators of densities, regression curves and modes. Ann. Stat.12, 766–774 (1984)

    Google Scholar 

  • Müller, H.G.: Weighted local regression and kernel methods for nonparametric curve fitting. J. Am. Stat. Assoc.82, 231–238 (1987)

    Google Scholar 

  • Müller, H.G.: On the construction of boundary kernels. University of California at Davis (Preprint 1991)

  • Nevai, P.: Some properties of orthogonal polynomials corresponding to the weight (1+x 2k)αexp(−x 2k) and their application in approximation theory. Sov. Math. Dokl.14, 1116–1119 (1973a)

    Google Scholar 

  • Nevai, P.: Orthogonal polynomials on the real line associated with the weight |x|αexp(−|x|β), I. Acta Math. Acad. Sci. Hung.24, 335–342 (1973b)

    Google Scholar 

  • Nevai, P.: Orthogonal polynomials. Mem. Am. Math. Soc.219 (1979)

  • Schucany, W.R.: On nonparametric regression with higher-order kernels. J. Stat. Plann. Inference23, 145–151 (1989)

    Google Scholar 

  • Schucany, W.R.: Sommers, J.P.: Improvement of kernel type density estimators. J. Am. Stat. Assoc.72, 420–423 (1977)

    Google Scholar 

  • Silverman, B.W.: Density estimation for statistics and data analysis. London: Chapman and Hall 1986

    Google Scholar 

  • Singh, R.S.: Mean squared errors of estimates of a density and its derivatives. Biometrika66(1), 177–180 (1979)

    Google Scholar 

  • Stuetzle, W., Mittal, Y.: Some comments on the asymptotic behavior of robust smoothers. In: Gasser, T., Rosenblatt, M. (eds.) Smoothing techniques for curve estimation. (Lect. Notes Math., vol. 757, pp. 191–195) Berlin Heidelberg New York: Springer 1979

    Google Scholar 

  • Wand, M., Schucany, W.R.: Gaussian-based kernels. Can. J. Stat.18, 197–204 (1990)

    Google Scholar 

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Authors and Affiliations

  1. Unité de Biométrie, ENSA.M, INRA, Montpellier II. 9, place Pierre Viala, F-34 060, Montpellier Cedex 1, France

    Alain Berlinet

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  1. Alain Berlinet
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Berlinet, A. Hierarchies of higher order kernels. Probab. Th. Rel. Fields 94, 489–504 (1993). https://doi.org/10.1007/BF01192560

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  • Received: 22 October 1991

  • Revised: 18 May 1992

  • Issue Date: December 1993

  • DOI: https://doi.org/10.1007/BF01192560

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Mathematics Subject Classification (1980)

  • 62 G 05
  • 62 G 20
  • 49 B 34
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