Least-squares trigonometric regression estimation

Popiński, Waldemar

Applicationes Mathematicae, Tome 26 (1999), p. 121-131 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $x_{i n} \in [0, 2 π]$ , i=1,...,n, are equidistant. Conditions for convergence of the mean-square prediction error $(1 / n) \sum_{i = 1}^{n} E {(f (x_{i n}) - {\hat{f}}_{N (n)} (x_{i n}))}^{2}$ , the integrated mean-square error $E ‖ f - {\hat{f}}_{N (n)} ‖^{2}$ and the pointwise mean-square error $E {(f (x) -_{N (n)} (x))}^{2}$ of the estimator ${\hat{f}}_{N (n)} (x) = \sum_{k = 0}^{N (n)} {\hat{c}}_{k} e_{k} (x)$ for f ∈ C[0,2π] and ${\hat{c}}_{0}, {\hat{c}}_{1}, . . ., {\hat{c}}_{N (n)}$ obtained by the least squares method are studied.

Publié le : 1999-01-01

Zbl 0992.62037

EUDML-ID : urn:eudml:doc:219229

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     author = {Waldemar Popi\'nski},
     title = {Least-squares trigonometric regression estimation},
     journal = {Applicationes Mathematicae},
     volume = {26},
     year = {1999},
     pages = {121-131},
     zbl = {0992.62037},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i2p121bwm}
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Popiński, Waldemar. Least-squares trigonometric regression estimation. Applicationes Mathematicae, Tome 26 (1999) pp. 121-131. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i2p121bwm/

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