Least-squares trigonometric regression estimation

• Volume: 26, Issue: 2, page 121-131
• ISSN: 1233-7234

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Abstract

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The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions ${e}_{k}$, k=0,1,2,..., for the observation model ${y}_{i}=f\left({x}_{in}\right)+{\eta }_{i}$, i=1,...,n, is considered, where ${\eta }_{i}$ are uncorrelated random variables with zero mean value and finite variance, and the observation points ${x}_{in}\in \left[0,2\pi \right]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $\left(1/n\right){\sum }_{i=1}^{n}E{\left(f\left({x}_{in}\right)-{\stackrel{^}{f}}_{N\left(n\right)}\left({x}_{in}\right)\right)}^{2}$, the integrated mean-square error $E‖f-{\stackrel{^}{f}}_{N\left(n\right)}{‖}^{2}$ and the pointwise mean-square error $E{\left(f\left(x\right){-}_{N\left(n\right)}\left(x\right)\right)}^{2}$ of the estimator ${\stackrel{^}{f}}_{N\left(n\right)}\left(x\right)={\sum }_{k=0}^{N\left(n\right)}{\stackrel{^}{c}}_{k}{e}_{k}\left(x\right)$ for f ∈ C[0,2π] and ${\stackrel{^}{c}}_{0},{\stackrel{^}{c}}_{1},...,{\stackrel{^}{c}}_{N\left(n\right)}$ obtained by the least squares method are studied.

How to cite

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Popiński, Waldemar. "Least-squares trigonometric regression estimation." Applicationes Mathematicae 26.2 (1999): 121-131. <http://eudml.org/doc/219229>.

@article{Popiński1999,
abstract = {The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i = f(x_\{in\}) + η_i$, i=1,...,n, is considered, where $η_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_\{in\} ∈ [0,2π]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n)\sum _\{i=1\}^n E(f(x_\{in\})-\widehat\{f\}_\{N(n)\}(x_\{in\}))^2$, the integrated mean-square error $E ‖f-\widehat\{f\}_\{N(n)\}‖^2$ and the pointwise mean-square error $E(f(x)-_\{N(n)\}(x))^2$ of the estimator $\widehat\{f\}_\{N(n)\}(x) = \sum _\{k=0\}^\{N(n)\} \widehat\{c\}_k e_k(x)$ for f ∈ C[0,2π] and $\widehat\{c\}_0,\widehat\{c\}_1,...,\widehat\{c\}_\{N(n)\}$ obtained by the least squares method are studied.},
author = {Popiński, Waldemar},
journal = {Applicationes Mathematicae},
keywords = {consistent estimator; least squares method; Fourier coefficients; trigonometric polynomial; regression function; trigonometric polynomials; least squares; consistent estimates},
language = {eng},
number = {2},
pages = {121-131},
title = {Least-squares trigonometric regression estimation},
url = {http://eudml.org/doc/219229},
volume = {26},
year = {1999},
}

TY - JOUR
AU - Popiński, Waldemar
TI - Least-squares trigonometric regression estimation
JO - Applicationes Mathematicae
PY - 1999
VL - 26
IS - 2
SP - 121
EP - 131
AB - The problem of nonparametric function fitting using the complete orthogonal system of trigonometric functions $e_k$, k=0,1,2,..., for the observation model $y_i = f(x_{in}) + η_i$, i=1,...,n, is considered, where $η_i$ are uncorrelated random variables with zero mean value and finite variance, and the observation points $x_{in} ∈ [0,2π]$, i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1/n)\sum _{i=1}^n E(f(x_{in})-\widehat{f}_{N(n)}(x_{in}))^2$, the integrated mean-square error $E ‖f-\widehat{f}_{N(n)}‖^2$ and the pointwise mean-square error $E(f(x)-_{N(n)}(x))^2$ of the estimator $\widehat{f}_{N(n)}(x) = \sum _{k=0}^{N(n)} \widehat{c}_k e_k(x)$ for f ∈ C[0,2π] and $\widehat{c}_0,\widehat{c}_1,...,\widehat{c}_{N(n)}$ obtained by the least squares method are studied.
LA - eng
KW - consistent estimator; least squares method; Fourier coefficients; trigonometric polynomial; regression function; trigonometric polynomials; least squares; consistent estimates
UR - http://eudml.org/doc/219229
ER -

References

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