# A note on orthogonal series regression function estimators

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

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## Abstract

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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 $\left({X}_{i},{Y}_{i}\right)$, 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 ${\stackrel{^}{f}}_{n}\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\stackrel{^}{c}}_{k}{e}_{k}\left(x\right)$, where the coefficients ${\stackrel{^}{c}}_{0},{\stackrel{^}{c}}_{1},...,{\stackrel{^}{c}}_{N}$ are determined by minimizing the empirical risk ${n}^{-1}{\sum }_{i=1}^{n}{\left({Y}_{i}-{\sum }_{k=0}^{N}{c}_{k}{e}_{k}\left({X}_{i}\right)\right)}^{2}$. Sufficient conditions for consistency of the estimators in the sense of the errors ${E}_{X}{|f\left(X\right)-{\stackrel{^}{f}}_{n}\left(X\right)|}^{2}$ and ${n}^{-1}{\sum }_{i=1}^{n}E{\left(f\left({X}_{i}\right)-{\stackrel{^}{f}}_{n}\left({X}_{i}\right)\right)}^{2}$ are obtained.

## How to cite

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Popiński, 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

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1. [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. [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. [3] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (1995), 677-687. Zbl0818.62041
4. [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. [5] --, Consistency of trigonometric and polynomial regression estimators, ibid. 25 (1998), 73-83.
6.
7. [7] V. N. Vapnik, Estimation of Dependencies Based on Empirical Data, Springer, New York, 1982. Zbl0499.62005

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