# Consistency of trigonometric and polynomial regression estimators

• Volume: 25, Issue: 1, page 73-83
• ISSN: 1233-7234

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## Abstract

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The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials ${e}_{k}$, k=0,1,..., for the observation model ${y}_{i}=f\left({x}_{i}\right)+{\eta }_{i}$, i=1,...,n, where the ${\eta }_{i}$ are independent random variables with zero mean value and finite variance, and the observation points ${x}_{i}\in \left[a,b\right]$, i=1,...,n, form a random sample from a distribution with density $\varrho \in {L}^{1}\left[a,b\right]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\parallel f-{\stackrel{^}{f}}_{N}\parallel ,|f\left(x\right){-}_{N}\left(x\right)|$, $x\in \left[a,b\right]$, and $E\parallel f{-}_{N}{\parallel }^{2}$ of the projection estimator ${\stackrel{^}{f}}_{N}\left(x\right)={\sum }_{k=0}^{N}{\stackrel{^}{c}}_{k}{e}_{k}\left(x\right)$ for ${\stackrel{^}{c}}_{0},{\stackrel{^}{c}}_{1},...,{\stackrel{^}{c}}_{N}$ determined by the least squares method and $f\in {L}^{2}\left[a,b\right]$.

## How to cite

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Popiński, Waldemar. "Consistency of trigonometric and polynomial regression estimators." Applicationes Mathematicae 25.1 (1998): 73-83. <http://eudml.org/doc/219195>.

@article{Popiński1998,
abstract = {The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i = f(x_i) + η_i$, i=1,...,n, where the $η_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in [a,b]$, i=1,...,n, form a random sample from a distribution with density $ϱ\in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\Vert f-\widehat\{f\}_N\Vert , \vert f(x)-_N(x)\vert$, $x\in [a,b]$, and $E\Vert f-_N\Vert ^2$ of the projection estimator $\widehat\{f\}_N(x) = \sum _\{k=0\}^N\widehat\{c\}_ke_k(x)$ for $\widehat\{c\}_0,\widehat\{c\}_1,\ldots ,\widehat\{c\}_N$ determined by the least squares method and $f\in L^2[a,b]$.},
author = {Popiński, Waldemar},
journal = {Applicationes Mathematicae},
keywords = {consistent estimator; orthonormal system; least squares method; regression; nonparametric regression},
language = {eng},
number = {1},
pages = {73-83},
title = {Consistency of trigonometric and polynomial regression estimators},
url = {http://eudml.org/doc/219195},
volume = {25},
year = {1998},
}

TY - JOUR
AU - Popiński, Waldemar
TI - Consistency of trigonometric and polynomial regression estimators
JO - Applicationes Mathematicae
PY - 1998
VL - 25
IS - 1
SP - 73
EP - 83
AB - The problem of nonparametric regression function estimation is considered using the complete orthonormal system of trigonometric functions or Legendre polynomials $e_k$, k=0,1,..., for the observation model $y_i = f(x_i) + η_i$, i=1,...,n, where the $η_i$ are independent random variables with zero mean value and finite variance, and the observation points $x_i\in [a,b]$, i=1,...,n, form a random sample from a distribution with density $ϱ\in L^1[a,b]$. Sufficient and necessary conditions are obtained for consistency in the sense of the errors $\Vert f-\widehat{f}_N\Vert , \vert f(x)-_N(x)\vert$, $x\in [a,b]$, and $E\Vert f-_N\Vert ^2$ of the projection estimator $\widehat{f}_N(x) = \sum _{k=0}^N\widehat{c}_ke_k(x)$ for $\widehat{c}_0,\widehat{c}_1,\ldots ,\widehat{c}_N$ determined by the least squares method and $f\in L^2[a,b]$.
LA - eng
KW - consistent estimator; orthonormal system; least squares method; regression; nonparametric regression
UR - http://eudml.org/doc/219195
ER -

## References

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6. [6] G. V. Milovanovič, D. S. Mitrinovič and T. M. Rassias, Topics on Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994. Zbl0848.26001
7. [7] W. Popiński, On least squares estimation of Fourier coefficients and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102. Zbl0789.62032
8. [8] W. Popiński, On Fourier coefficient estimators consistent in the mean-square sense, ibid., 275-284. Zbl0801.62040
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10. [10] G. Sansone, Orthogonal Functions, Interscience Publ., New York, 1959. Zbl0084.06106

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