Least-squares trigonometric regression estimation

Waldemar Popiński

Applicationes Mathematicae (1999)

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

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 ( x i n ) + 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 i n ∈ [ 0 , 2 π ] , i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error ( 1 / n ) ∑ i = 1 n E ( f ( x i n ) - f ^ N ( n ) ( x i n ) ) 2 , the integrated mean-square error E ‖ f - f ^ N ( n ) ‖ 2 and the pointwise mean-square error E ( f ( x ) - N ( n ) ( x ) ) 2 of the estimator f ^ N ( n ) ( x ) = ∑ k = 0 N ( n ) c ^ k e k ( x ) for f ∈ C[0,2π] and c ^ 0 , c ^ 1 , . . . , c ^ N ( n ) 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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  1. [1] B. Droge, On finite-sample properties of adaptive least-squares regression estimates, Statistics 24 (1993), 181-203. Zbl0808.62035
  2. [2] R. L. Eubank and P. Speckman, Convergence rates for trigonometric and polynomial-trigonometric regression estimators, Statist. Probab. Lett. 11 (1991), 119-124. Zbl0712.62037
  3. [3] T. Gasser, L. Sroka and C. Jennen-Steinmetz, Residual variance and residual pattern in nonlinear regression, Biometrika 73 (1986), 625-633. Zbl0649.62035
  4. [4] P. Hall, J. W. Kay and D. M. Titterington, Asymptotically optimal difference-based estimation of variance in nonparametric regression, ibid. 77 (1990), 521-528. 
  5. [5] P. Hall and P. Patil, On wavelet methods for estimating smooth functions, J. Bernoulli Soc. 1 (1995), 41-58. Zbl0830.62037
  6. [6] G. G. Lorentz, Approximation of Functions, Holt, Reinehart & Winston, New York, 1966. Zbl0153.38901
  7. [7] C. L. Mallows, Some comments on C p , Technometrics 15 (1973), 661-675. 
  8. [8] E. Nadaraya, Limit distribution of the integrated squared error of trigonometric series regression estimator, Proc. Georgian Acad. Sci. Math. 1 (1993), 221-237. Zbl0796.62039
  9. [9] B. T. Polyak and A. B. Tsybakov, Asymptotic optimality of the C p criterion in projection type estimation of regression functions, Teor. Veroyatnost. Primenen. 35 (1990), 305-317 (in Russian). 
  10. [10] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358. Zbl0649.62034
  11. [11] A. Zygmund, Trigonometrical Series, Dover, 1955. Zbl0065.05604

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