Annals of the Institute of Statistical Mathematics

, Volume 35, Issue 1, pp 215–228

An orthogonal series estimate of time-varying regression

  • Wŀodzimierz Greblicki
  • Danuta Rutkowska
  • Leszek Rutkowski
Article

Summary

Let (X1,Y1), (X2,Y2),... be independent pairs of random variables according to the modelYn=tn(Xn)R(Xn)+Zn,n=1,2,..., wheretn andR are unknown functions.Zn's are i.i.d. random variables with zero mean and finite variance. The marginal density ofXn is independent ofn. In the paper nonparametric estimates of a nonstationary regression function E{Yn|Xn=x}=tn(x)R(x) are proposed and their asymptotic properties are investigated.

Key words and phrases

Nonparametrio estimation regression function nonstationary regression function orthogonal series stochastic approximation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bosq, D. (1969). Sur l'estimation d'une densite multivariee par une serie de fonctions orthogonales,C. R. Acad. Sci. Paris, A-B,268, 555–557.MATHMathSciNetGoogle Scholar
  2. [2]
    Ĉencov, N. N. (1962). Evaluation of an unknown distribution density from observations,Soviet Math.,3, 1559–1562.Google Scholar
  3. [3]
    Chung, K. L. (1954). On a stochastic approximation method.Ann. Math. Statist.,25, 463–483.MATHGoogle Scholar
  4. [4]
    Collomb, G. (1977). Quelques proprietes de la methode du noyau pour l'estimation non parametrique de la regression en un point fixe.C. R. Acad. Sci. Paris, A,285, 289–292.MATHMathSciNetGoogle Scholar
  5. [5]
    Devroye, L. P. (1978). The uniform convergence of nearest neighbor regression function estimators and their application in optimization,IEEE Trans. Inf. Theory,24, 142–151.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Devroye, L. P. and Wagner, T. J. (1980). On theL 1 convergence of kernel regression function estimators with applications in discrimination.Zeit. Wahrscheinlichkeitsth.,51, 15–25.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Doob, J. L. (1953).Stochastic Processes, John Wiley, New York.MATHGoogle Scholar
  8. [8]
    Dupaĉ, V. (1965). A dynamic stochastic approximation method,Ann. Math. Statist.,36, 1695–1702.MathSciNetGoogle Scholar
  9. [9]
    Greblicki, W. (1974). Asymptotically optimal probabilistic algorithms for pattern recognition and identification, Scientific Papers of the Institute of Technical Cybernetics of Wroclaw Technical University No. 18, Series: Monographs No. 3, Wroclaw (in Polish).Google Scholar
  10. [10]
    Greblicki, W. and Krzyżak, A. (1980). Asymptotic properties of kernel estimates of a regression function.J. Statist. Plann. Inf.,4, 81–90.MATHCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    Mirzahmedov, M. A. and Haŝimov, S. A. (1972). On some properties of density estimation,Colloq. Math. Soc. Janos Bolyai, European meeting of statisticians, Budapest, 535–546.Google Scholar
  13. [13]
    Nadaraya, E. A. (1964). On estimating regression,Theory Prob. Appl.,9, 141–142.CrossRefGoogle Scholar
  14. [14]
    Noda, K. (1976). Estimation of a regression function by the Parzen kernel-type density estimators,Ann. Inst. Statist. Math.,28, A, 221–234.MATHMathSciNetGoogle Scholar
  15. [15]
    Rosenblatt, M. (1969). Conditional probability density and regression estimators,Multivariate Analysis, II, Academic Press, New York, 25–31.Google Scholar
  16. [16]
    Sansone, G. (1959).Orthogonal Functions, Interscience Publishers Inc., New York.MATHGoogle Scholar
  17. [17]
    Schwartz, S. C. (1967). Estimation of probability density by an orthogonal series,Ann. Math. Statist.,38, 1261–1265.MATHMathSciNetGoogle Scholar
  18. [18]
    Stone, C. J. (1977). Consistent nonparametric regression,Ann. Statist.,5, 595–645.MATHMathSciNetGoogle Scholar
  19. [19]
    Szegö, G. (1959).Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., 23.Google Scholar
  20. [20]
    Van Ryzin, J. (1969). On strong consistency of density estimates,Ann. Math. Statist.,40, 1765–1772.MATHMathSciNetGoogle Scholar
  21. [21]
    Wasan, M. T. (1969).Stochastic Approximation, Cambridge University Press.Google Scholar

Copyright information

© Kluwer Academic Publishers 1983

Authors and Affiliations

  • Wŀodzimierz Greblicki
    • 1
    • 2
    • 3
  • Danuta Rutkowska
    • 1
    • 2
    • 3
  • Leszek Rutkowski
    • 1
    • 2
    • 3
  1. 1.Technical University of WrocŀawWrocŀawPoland
  2. 2.Technical University of CzęstochowaCzęstochowaPoland
  3. 3.Technical University of CzęstochowaCzęstochowaPoland

Personalised recommendations