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Tytuł artykułu
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Warianty tytułu
Języki publikacji
Abstrakty
The problem of nonparametric function fitting with the observation model $y_i = f(x_i) + η_i$, i=1,...,n, is considered, where $η_i$ are independent random variables with zero mean value and finite variance, and $x_i \in [a,b] \subset \R^1$, i=1,...,n, form a random sample from a distribution with density $ϱ \in L^1[a,b]$ and are independent of the errors $η_i$, i=1,...,n. The asymptotic properties of the estimator $\widehat{f}_{N(n)}(x) = \sum_{k=1}^{N(n)} \widehat{c}_ke_k(x)$ for $f \in L^2[a,b]$ and $\widehat{c}^{N(n)}=( \widehat{c}_1,..., \widehat{c}_{N(n)})^T$ obtained by the least squares method as well as the limits in probability of the estimators $\widehat{c}_k$, k=1,...,N, for fixed N, are studied in the case when the functions $e_k$, k=1,2,..., forming a complete orthonormal system in $L^2\[a,b\]$ are analytic.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
91-102
Opis fizyczny
Daty
wydano
1993
otrzymano
1992-12-1
Twórcy
autor
- Research and Development Center of Statistics, Al. Niepodległości 208, 00-925 Warszawa, Poland
Bibliografia
- [1] H. Akaike, A new look at the statistical model identification, IEEE Trans. Automat. Control AC-19 (1974), 716-723
- [2] Y. S. Chow and H. Teicher, Probability Theory, Independence, Interchangeability, Martingales, Springer, Heidelberg, 1978
- [3] C. L. Mallows, Some comments on C_p, Technometrics 15 (1973), 661-675
- [4] B. T. Polyak and A. B. Tsybakov, Asymptotic optimality of the C_p criterion in projection type estimation of a regression function, Teor. Veroyatnost. i Primenen. 35 (1990), 305-317 (in Russian)
- [5] E. Rafajłowicz, Nonparametric least-squares estimation of a regression function, Statistics 19 (1988), 349-358
- [6] G. Sansone, Orthogonal Functions, Interscience, New York, 1959.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
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