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

Applicationes Mathematicae (1994)

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

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topPopiński, 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

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

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