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## Applicationes Mathematicae

1998-1999 | 25 | 1 | 73-83
Tytuł artykułu

### Consistency of trigonometric and polynomial regression estimators

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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 $\Vert f-\widehat f_N\Vert, \vert f(x)-\widehatf_N(x)\vert$, $x\in[a,b]$, and $E\Vert f-\widehatf_N\Vert^2$ 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 $f\in L^2[a,b]$.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
73-83
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-01-22
poprawiono
1997-05-23
Twórcy
autor
• Department of Standards, Central Statistical Office, Al. Niepodległości 208, 00-925 Warszawa, Poland
Bibliografia
• [1] R. L. Eubank and B. R. Jayasuriya, The asymptotic average square error for polynomial regression, Statistics 24 (1993), 311-319.
• [2] J. Fan and I. Gijbels, Local Polynomial Modelling and Its Applications, Chapman and Hall, London, 1996.
• [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] G. G. Lorentz, Approximation of Functions, Holt, Reinehart and Winston, New York, 1966.
• [5] G. Lugosi and K. Zeger, Nonparametric estimation via empirical risk minimization, IEEE Trans. Inform. Theory IT-41 (3) (1995), 677-687.
• [6] G. V. Milovanovič, D. S. Mitrinovič and T. M. Rassias, Topics on Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994.
• [7] W. Popiński, On least squares estimation of Fourier coefficients and of the regression function, Appl. Math. (Warsaw) 22 (1993), 91-102.
• [8] W. Popiński, On Fourier coefficient estimators consistent in the mean-square sense, ibid., 275-284.
• [9] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358.
• [10] G. Sansone, Orthogonal Functions, Interscience Publ., New York, 1959.
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Bibliografia
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