Convergence rates of orthogonal series regression estimators

Waldemar Popiński

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 4, page 445-454
  • ISSN: 1233-7234

Abstract

top
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 (Yi,Xi), 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 ) - f ^ N ( X ) ) 2 and ∥ f - f ^ N ∥ ∞ for the estimator f ^ N ( x ) = ∑ k = 1 N c ^ k e k ( x ) , constructed using an orthonormal system e k , k=1,2,..., in L 2 ( A ) are obtained.

How to cite

top

Popiński, Waldemar. "Convergence rates of orthogonal series regression estimators." Applicationes Mathematicae 27.4 (2000): 445-454. <http://eudml.org/doc/219287>.

@article{Popiński2000,
abstract = {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 (Yi,Xi), 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.},
author = {Popiński, Waldemar},
journal = {Applicationes Mathematicae},
keywords = {orthonormal system; nonparametric series regression; least squares method; convergence rate},
language = {eng},
number = {4},
pages = {445-454},
title = {Convergence rates of orthogonal series regression estimators},
url = {http://eudml.org/doc/219287},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Popiński, Waldemar
TI - Convergence rates of orthogonal series regression estimators
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 4
SP - 445
EP - 454
AB - 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 (Yi,Xi), 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.
LA - eng
KW - orthonormal system; nonparametric series regression; least squares method; convergence rate
UR - http://eudml.org/doc/219287
ER -

References

top
  1. [1] L. Birgé and P. Massart, Minimum contrast estimators on sieves: exponential bounds and rates of convergence, J. Bernoulli Soc. 4 (1998), 329-375. Zbl0954.62033
  2. [2] D. D. Cox, Approximation of least squares regression on nested subspaces, Ann. Statist. 16 (1988), 713-732. Zbl0669.62047
  3. [3] L. Györfi and H. Walk, On the strong universal consistency of a series type regression estimate, Math. Methods Statist. 5 (1996), 332-342. Zbl0874.62048
  4. [4] J. Z. Huang, Projection estimation in multiple regression with application to functional ANOVA models, Ann. Statist. 26 (1998), 242-272. Zbl0930.62042
  5. [5] G. G. Lorentz, Approximation of Functions, Holt, Reinehart & Winston, New York, 1966. Zbl0153.38901
  6. [6] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (3) (1995), 677-687. Zbl0818.62041
  7. [7] P. Niyogi and F. Girosi, Generalization bounds for function approximation from scattered noisy data, Adv. Comput. Math. 10 (1999), 51-80. Zbl1053.65506
  8. [8] W. Popiński, On least squares estimation of Fourier coefficients , and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102. Zbl0789.62032
  9. [9] W. Popiński, Consistency of trigonometric and polynomial regression estimators, ibid. 25 (1998), 73-83. Zbl0895.62047
  10. [10] W. Popiński, A note on orthogonal series regression function estimators, ibid. 26 (1999), 281-291. Zbl0992.62039
  11. [11] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358. Zbl0649.62034
  12. [12] C. J. Stone, Optimal global rates of convergence for nonparametric regression, Ann. Statist. 10 (1982), 1040-1053. Zbl0511.62048
  13. [13] G. Viennet, Least-square estimation for regression on random design for absolutely regular observations, Statist. Probab. Lett. 43 (1999), 13-23. Zbl0933.62034

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.