Advertisement

Sequential Estimates of a Regression Function by Orthogonal Series with Applications in Discrimination

  • Leszek Rutkowski
Part of the Lecture Notes in Statistics book series (LNS, volume 8)

Abstract

Let (X,Y) be a pair of random variables. X takes values in a Borel set A, A⊂ Rp, whereas Y takes values in R. Let f be the marginal Leb. esgue density of X. Based on a sample (X1, Y1),…, (Xn, Yn) of independent observations of (X,Y) we wish to estimate the regression r of Y on X, i.e
$${\rm{r(x) = E[Y|X = x]}}{\rm{.}}$$

Keywords

Regression Function Complete Orthonormal System Recursive Version Universal Consistency Pattern Recognition Procedure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [l]
    Ahmad, I. A. and Lin, P. E., “Nonparametric sequential estimation of a multiple regression function,” Bull. Math. Statist., vol. 17, pp. 63–75, 1976.MathSciNetMATHGoogle Scholar
  2. [2]
    Čencov, N. N., “Evaluation of an unknown distribution density from observations,” Soviet Math., vol. 3, pp. 1559–1562, 1962.Google Scholar
  3. [3]
    Devroye, L. P., “Universal consistency in nonparametric regression and nonparametric discrimination,” Technical Report School of Computer Science, McGill university, 1978.Google Scholar
  4. [4]
    Devroye, 1. P. and Wagner, T. J., “On the L1 convergence of kernel estimators of regression functions with applications in discrimination,” to appear in Z. Wahrscheinlichkeitstheorie und Verw. Gebiete.Google Scholar
  5. [5]
    Greblicki, W., “Asymptotically optimal probabilistic algorithms for pattern recognition and identification,” Scientific Papers of the Institute of Technical Cybernetics of Wroclaw Technical University Wo. 18, Series: Monographs No. 3, Wroclaw 1974.Google Scholar
  6. [6]
    Greblicki, W., “Asymptotically optimal pattern recognition procedures with density estimates,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 250–251, 1978.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Loéve, M., “Probability Theory I,” 4th Edition, Springer-Verlag, 1977.Google Scholar
  8. [8]
    Sansone, G., “Orthogonal functions,” Interscience Publishers Inc., New York, 1959.MATHGoogle Scholar
  9. [9]
    Szegö, G., “Orthogonal polynomials,” Amer. Math. Soc. Coll. Publ., vol. 23, 1959.Google Scholar
  10. [10]
    Tucker, H. G., “A graduate course in probability,” Academic Press, 1967.Google Scholar
  11. [l1]
    Wertz, W. and Schneider, B., “Statistical density estimation: a bibliography,” Intexnat. Statist. Rev., vol. 47, pp. 155–175, 1979.MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1981

Authors and Affiliations

  • Leszek Rutkowski
    • 1
  1. 1.Technical University of CzęstochowaPoland

Personalised recommendations