Consistency of trigonometric and polynomial regression estimators

Waldemar Popiński

Applicationes Mathematicae (1998)

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

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 ( 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 ∈ [ a , b ] , i=1,...,n, form a random sample from a distribution with density ϱ ∈ L 1 [ a , b ] . Sufficient and necessary conditions are obtained for consistency in the sense of the errors ∥ f - f ^ N ∥ , | f ( x ) - N ( x ) | , x ∈ [ a , b ] , and E ∥ f - N ∥ 2 of the projection estimator f ^ N ( x ) = ∑ k = 0 N c ^ k e k ( x ) for c ^ 0 , c ^ 1 , ... , c ^ N determined by the least squares method and f ∈ L 2 [ a , b ] .

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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  1. [1] R. L. Eubank and B. R. Jayasuriya, The asymptotic average square error for polynomial regression, Statistics 24 (1993), 311-319. Zbl0808.62036
  2. [2] J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman and Hall, London, 1996. 
  3. [3] A. R. Gallant and H. White, There exists a neural network that does not make avoidable mistakes, in: Proc. Second Annual IEEE Conf. on Neural Networks, San Diego, Calif., IEEE Press, New York, 1988, 657-664. 
  4. [4] G. G. Lorentz, Approximation of Functions, Holt, Reinehart and Winston, New York, 1966. 
  5. [5] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (3) (1995), 677-687. Zbl0818.62041
  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
  9. [9] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358. Zbl0649.62034
  10. [10] G. Sansone, Orthogonal Functions, Interscience Publ., New York, 1959. Zbl0084.06106

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