On least squares estimation of Fourier coefficients and of the regression function

• Autorzy: Waldemar Popiński
• Języki publikacji: EN

Abstrakty

EN

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

EN

Fourier series; consistent estimator; least squares method; regression

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1993-1995
• Tom: 22
• Numer: 1
• Strony: 91-102

Daty

wydano

1993

otrzymano

1992-12-1

Twórcy

autor

Waldemar Popiński

• Research and Development Center of Statistics, Al. Niepodległości 208, 00-925 Warszawa, Poland

Bibliografia

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• [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.