Consistency of trigonometric and polynomial regression estimators

Popiński, Waldemar

Applicationes Mathematicae, Tome 25 (1998), p. 73-83 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $∥ f - {\hat{f}}_{N} ∥, | f (x) -_{N} (x) |$ , $x \in [a, b]$ , and $E ∥ f -_{N} ∥^{2}$ of the projection estimator ${\hat{f}}_{N} (x) = \sum_{k = 0}^{N} {\hat{c}}_{k} e_{k} (x)$ for ${\hat{c}}_{0}, {\hat{c}}_{1}, ..., {\hat{c}}_{N}$ determined by the least squares method and $f \in L^{2} [a, b]$ .

Publié le : 1998-01-01

Zbl 0895.62047

EUDML-ID : urn:eudml:doc:219195

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     author = {Waldemar Popi\'nski},
     title = {Consistency of trigonometric and polynomial regression estimators},
     journal = {Applicationes Mathematicae},
     volume = {25},
     year = {1998},
     pages = {73-83},
     zbl = {0895.62047},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv25i1p73bwm}
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Popiński, Waldemar. Consistency of trigonometric and polynomial regression estimators. Applicationes Mathematicae, Tome 25 (1998) pp. 73-83. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv25i1p73bwm/

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