# On Fourier coefficient estimators consistent in the mean-square sense

• Volume: 22, Issue: 2, page 275-284
• ISSN: 1233-7234

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## Abstract

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The properties of two recursive estimators of the Fourier coefficients of a regression function $f\in {L}^{2}\left[a,b\right]$ with respect to a complete orthonormal system of bounded functions (ek) , k=1,2,..., are considered in the case of the observation model ${y}_{i}=f\left({x}_{i}\right)+{\eta }_{i}$, i=1,...,n , where ${\eta }_{i}$ are independent random variables with zero mean and finite variance, ${x}_{i}\in \left[a,b\right]\subset {R}^{1}$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors ${\eta }_{i}$, i=1,...,n . Unbiasedness and mean-square consistency of the examined estimators are proved and their mean-square errors are compared.

## How to cite

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Popiński, Waldemar. "On Fourier coefficient estimators consistent in the mean-square sense." Applicationes Mathematicae 22.2 (1994): 275-284. <http://eudml.org/doc/219095>.

@article{Popiński1994,
abstract = {The properties of two recursive estimators of the Fourier coefficients of a regression function $f \in L^2[a,b]$ with respect to a complete orthonormal system of bounded functions (ek) , k=1,2,..., are considered in the case of the observation model $y_i = f(x_i) + η_i$, i=1,...,n , where $η_i$ are independent random variables with zero mean and finite variance, $x_i \in [a,b] \subset \{R\}^1$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors $η_i$, i=1,...,n . Unbiasedness and mean-square consistency of the examined estimators are proved and their mean-square errors are compared.},
author = {Popiński, Waldemar},
journal = {Applicationes Mathematicae},
keywords = {unbiasedness; consistent estimator; Fourier coefficients; mean-square error; uniform distribution; recursive estimators; regression function; complete orthonormal system of bounded functions; mean-square consistency; mean-square errors},
language = {eng},
number = {2},
pages = {275-284},
title = {On Fourier coefficient estimators consistent in the mean-square sense},
url = {http://eudml.org/doc/219095},
volume = {22},
year = {1994},
}

TY - JOUR
AU - Popiński, Waldemar
TI - On Fourier coefficient estimators consistent in the mean-square sense
JO - Applicationes Mathematicae
PY - 1994
VL - 22
IS - 2
SP - 275
EP - 284
AB - The properties of two recursive estimators of the Fourier coefficients of a regression function $f \in L^2[a,b]$ with respect to a complete orthonormal system of bounded functions (ek) , k=1,2,..., are considered in the case of the observation model $y_i = f(x_i) + η_i$, i=1,...,n , where $η_i$ are independent random variables with zero mean and finite variance, $x_i \in [a,b] \subset {R}^1$, i=1,...,n, form a random sample from a distribution with density ϱ =1/(b-a) (uniform distribution) and are independent of the errors $η_i$, i=1,...,n . Unbiasedness and mean-square consistency of the examined estimators are proved and their mean-square errors are compared.
LA - eng
KW - unbiasedness; consistent estimator; Fourier coefficients; mean-square error; uniform distribution; recursive estimators; regression function; complete orthonormal system of bounded functions; mean-square consistency; mean-square errors
UR - http://eudml.org/doc/219095
ER -

## References

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1. [1] A. E. Albert and L. A. Gardner, Stochastic Approximation and Nonlinear Regression, Cambridge Univ. Press, 1967. Zbl0162.21502
2. [2] J. Koronacki, Stochastic Approximation-Optimization Methods under Random Conditions, WNT, Warszawa, 1989 (in Polish). Zbl0698.62084
3. [3] E. A. Nadaraya, Nonparametric Estimation of Probability Densities and Regression Curves, Kluwer Acad. Publ., Dordrecht, 1989. Zbl0709.62039
4. [4] G. Sansone, Orthogonal Functions, Interscience, New York, 1959.
5. [5] A. Zygmund, Trigonometrical Series, Dover, 1955. Zbl0065.05604

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