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2000 | 27 | 4 | 445-454
Tytuł artykułu

Convergence rates of orthogonal series regression estimators

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
General conditions for convergence rates of nonparametric orthogonal series estimators of the regression function f(x)=E(Y | X = x) are considered. The estimators are obtained by the least squares method on the basis of a random observation sample (Y_i,X_i), i=1,...,n, where $X_i ∈ A ⊂ ℝ^d$ have marginal distribution with density $ϱ ∈ L^1(A)$ and Var( Y | X = x) is bounded on A. Convergence rates of the errors $E_X(f(X)-\widehat f_N(X))^2$ and $\Vert f-\widehat f_N\Vert_∞$ for the estimator $\widehat f_N(x) = \sum_{k=1}^N\widehat c_ke_k(x)$, constructed using an orthonormal system $e_k$, k=1,2,..., in $L^2(A)$ are obtained.
Rocznik
Tom
27
Numer
4
Strony
445-454
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-12-23
poprawiono
2000-07-31
Twórcy
  • Department of Survey Organization, Central Statistical Office, Al. Niepodległości 208, 00-925 Warszawa, Poland
Bibliografia
  • [1] L. Birgé and P. Massart, Minimum contrast estimators on sieves: exponential bounds and rates of convergence, J. Bernoulli Soc. 4 (1998), 329-375.
  • [2] D. D. Cox, Approximation of least squares regression on nested subspaces, Ann. Statist. 16 (1988), 713-732.
  • [3] L. Györfi and H. Walk, On the strong universal consistency of a series type regression estimate, Math. Methods Statist. 5 (1996), 332-342.
  • [4] J. Z. Huang, Projection estimation in multiple regression with application to functional ANOVA models, Ann. Statist. 26 (1998), 242-272.
  • [5] G. G. Lorentz, Approximation of Functions, Holt, Reinehart & Winston, New York, 1966.
  • [6] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (3) (1995), 677-687.
  • [7] P. Niyogi and F. Girosi, Generalization bounds for function approximation from scattered noisy data, Adv. Comput. Math. 10 (1999), 51-80.
  • [8] W. Popiński, On least squares estimation of Fourier coefficients$,$ and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102.
  • [9] W. Popiński, Consistency of trigonometric and polynomial regression estimators, ibid. 25 (1998), 73-83.
  • [10] W. Popiński, A note on orthogonal series regression function estimators, ibid. 26 (1999), 281-291.
  • [11] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358.
  • [12] C. J. Stone, Optimal global rates of convergence for nonparametric regression, Ann. Statist. 10 (1982), 1040-1053.
  • [13] G. Viennet, Least-square estimation for regression on random design for absolutely regular observations, Statist. Probab. Lett. 43 (1999), 13-23.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-zmv27i4p445bwm
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