Fourier and Hermite series estimates of regression functions

  • Wlodzimier Greblicki
  • Miroslaw Pawlak


In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X 1,Y 1),…, (X n, Yn) of a pair (X, Y) of random variables. We examine an estimate of a type\({{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} \mathord{\left/ {\vphantom {{\hat m\left( x \right) = \sum\limits_{j = 1}^n {Y_{j\varphi N} } \left( {x,X_j } \right)} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j = 1}^n {\varphi _N } \left( {x,X_j } \right)}}\), whereN depends onn andϕ N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for\(\hat m\left( x \right)\) to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then\(\hat m\left( x \right)\) converges tom(x) as rapidly asO(nC−(2s−1)/4s) in probability andO(n −(2s−1)/4s logn) almost completely.

Key words and phrases

Regression function Fourier series Hermite series nonparameteric estimate 


  1. [1]
    Bluez, J. and Bosq, D. (1976). Conditions nécessaires et suffisantes de convergence pour une class d'estimateurs de la densité,C. R. Acad. Sci., Paris,282, 636–666.Google Scholar
  2. [2]
    Collomb, G. (1981). Estimation non paramétrique de la regression: revue bibliographique,Internat. Statist. Rev.,49, 75–93.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    Földes, A. and Révész, R. (1974). A general method for density estimation,Studia Sci. Math. Hungar.,9, 81–92.Google Scholar
  4. [4]
    Glick, N. (1974). Consistency conditions for probability estimators and integrals for density estimators,Utilitas Math.,5, 61–74.MathSciNetGoogle Scholar
  5. [5]
    Greblicki, W., Krzyzak, A. and Pawlak, M. (1984). Distribution-free pointwise consistency of kernel regression estimates,Ann. Statist.,12, 1570–1575.MATHMathSciNetGoogle Scholar
  6. [6]
    Greblicki, W. and Pawlak, M. (1984). Hermite series estimate of a probability density and its derivatives,J. Multivariate Anal.,15, 174–182.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Greblicki, W. and Pawlak, M. (1985). Pointwise consistency of the Hermite series density estimate,Probability and Statistics Letters, to appear.Google Scholar
  8. [8]
    Hoeffding, W. (1953). Probability inequalities for sums of bounded random variables,J. Amer. Statist. Ass.,58, 13–30.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Hunt, R. A. (1968). On the convergence of Fourier series, Orthogonal Expansion and their Continuous Analogues,Proc. Conf. Edwardsville, 1967, 235–255, Southern Illinois Univ. Press, Carbondale, Ill.Google Scholar
  10. [10]
    Kronmal, R. and Tarter, M. (1968). The estimation of probability densities and cumulatives by Fourier series methods,J. Amer. Statist. Ass.,63, 925–952.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Krzyzak, A. and Pawlak, M. (1984). Distribution-free consistency of nonparametric kernel regression estimate and classification,IEEE Trans. Inform. Theory, IT-30, 78–81.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    Nadaraya, E. A. (1964). On estimating regression,Theory of Probability and its Applications,9, 141–142.CrossRefGoogle Scholar
  13. [13]
    Sansone, G. (1959).Orthogonal Functions, Interscience Publishers Inc.Google Scholar
  14. [14]
    Schwartz, S. C. (1967). Estimation of probability density by an orthogonal series,Ann. Math. Statist.,38, 1261–1265.MATHMathSciNetGoogle Scholar
  15. [15]
    Stone, C. J. (1980). Optimal rates of convergence for non-parametric estimators,Ann. Statist.,8, 1348–1360.MATHMathSciNetGoogle Scholar
  16. [16]
    Szegö, G. (1959). Orthogonal Polynomials,Amer. Math. Soc. Coll. Publ. Google Scholar
  17. [17]
    Tapia, R. A. and Thompson, J. R. (1978).Nonparametric Probability Density Estimation, The John Hopkins University Press, Baltimore.MATHGoogle Scholar
  18. [18]
    Tarter, M. and Kronmal, R. (1970). On multivariate density estimates based on orthogonal expansions,Ann. Math. Statist.,41, 718–722.MATHMathSciNetGoogle Scholar
  19. [19]
    Wahba, G. (1975). Optimal convergence properties of varaible knot, kernel and orthogonal series methods for density estimation,Ann. Statist.,3, 15–29.MATHMathSciNetGoogle Scholar
  20. [20]
    Walter, G. G. (1977). Properties of Hermite series estimation of probability density,Ann. Statist.,5, 1258–1264.MATHMathSciNetGoogle Scholar
  21. [21]
    Watson, G. S. (1964). Smooth regression analysis,Sankhya, A,26, 359–372.MATHGoogle Scholar
  22. [22]
    Zygmund, A. (1959).Trigonometric series II, Cambridge University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1985

Authors and Affiliations

  • Wlodzimier Greblicki
    • 1
    • 2
  • Miroslaw Pawlak
    • 1
    • 2
  1. 1.Institute of Engineering CyberneticsTechnical University of WroclawWroclawPoland
  2. 2.Concordia UniversitymontrealCanada

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