# A note on orthogonal series regression function estimators

Applicationes Mathematicae (1999)

- Volume: 26, Issue: 3, page 281-291
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topPopiński, Waldemar. "A note on orthogonal series regression function estimators." Applicationes Mathematicae 26.3 (1999): 281-291. <http://eudml.org/doc/219239>.

@article{Popiński1999,

abstract = {The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_i,Y_i)$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^1$[a,b]. The constructed estimators are of the form $\widehat\{f\}_n(x) = \sum _\{k=0\}^\{N(n)\}\widehat\{c\}_ke_k(x)$, where the coefficients $\widehat\{c\}_0,\widehat\{c\}_1,...,\widehat\{c\}_N$ are determined by minimizing the empirical risk $n^\{-1\}\sum _\{i=1\}^n(Y_i - \sum _\{k=0\}^Nc_ke_k(X_i))^2$. Sufficient conditions for consistency of the estimators in the sense of the errors $E_X\vert f(X)-\widehat\{f\}_n(X)\vert ^2$ and $n^\{-1\}\sum _\{i=1\}^nE(f(X_i)-\widehat\{f\}_n(X_i))^2$ are obtained.},

author = {Popiński, Waldemar},

journal = {Applicationes Mathematicae},

keywords = {consistent estimator; orthonormal system; empirical risk minimization; nonparametric regression},

language = {eng},

number = {3},

pages = {281-291},

title = {A note on orthogonal series regression function estimators},

url = {http://eudml.org/doc/219239},

volume = {26},

year = {1999},

}

TY - JOUR

AU - Popiński, Waldemar

TI - A note on orthogonal series regression function estimators

JO - Applicationes Mathematicae

PY - 1999

VL - 26

IS - 3

SP - 281

EP - 291

AB - The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_i,Y_i)$, i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^1$[a,b]. The constructed estimators are of the form $\widehat{f}_n(x) = \sum _{k=0}^{N(n)}\widehat{c}_ke_k(x)$, where the coefficients $\widehat{c}_0,\widehat{c}_1,...,\widehat{c}_N$ are determined by minimizing the empirical risk $n^{-1}\sum _{i=1}^n(Y_i - \sum _{k=0}^Nc_ke_k(X_i))^2$. Sufficient conditions for consistency of the estimators in the sense of the errors $E_X\vert f(X)-\widehat{f}_n(X)\vert ^2$ and $n^{-1}\sum _{i=1}^nE(f(X_i)-\widehat{f}_n(X_i))^2$ are obtained.

LA - eng

KW - consistent estimator; orthonormal system; empirical risk minimization; nonparametric regression

UR - http://eudml.org/doc/219239

ER -

## References

top- [1] A. R. Gallant and H. White, There exists a neural network that does not make avoidable mistakes, in: Proc. Second Annual IEEE Conference on Neural Networks, San Diego, California, IEEE Press, New York, 1988, 657-664.
- [2] 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
- [3] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (1995), 677-687. Zbl0818.62041
- [4] W. Popiński, On least squares estimation of Fourier coefficients and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102. Zbl0789.62032
- [5] --, Consistency of trigonometric and polynomial regression estimators, ibid. 25 (1998), 73-83.
- [7] V. N. Vapnik, Estimation of Dependencies Based on Empirical Data, Springer, New York, 1982. Zbl0499.62005

## Citations in EuDML Documents

top## NotesEmbed ?

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