A note on orthogonal series regression function estimators

Popiński, Waldemar

Applicationes Mathematicae, Tome 26 (1999), p. 281-291 / Harvested from The Polish Digital Mathematics Library

Résumé

The problem of nonparametric estimation of the regression function f(x) = E(Y | X=x) using the orthonormal system of trigonometric functions or Legendre polynomials $e_{k}$ , k=0,1,2,..., is considered in the case where a sample of i.i.d. copies $(X_{i}, Y_{i})$ , i=1,...,n, of the random variable (X,Y) is available and the marginal distribution of X has density ϱ ∈ $L^{1}$ [a,b]. The constructed estimators are of the form ${\hat{f}}_{n} (x) = \sum_{k = 0}^{N (n)} {\hat{c}}_{k} e_{k} (x)$ , where the coefficients ${\hat{c}}_{0}, {\hat{c}}_{1}, . . ., {\hat{c}}_{N}$ are determined by minimizing the empirical risk $n^{- 1} \sum_{i = 1}^{n} {(Y_{i} - \sum_{k = 0}^{N} c_{k} e_{k} (X_{i}))}^{2}$ . Sufficient conditions for consistency of the estimators in the sense of the errors $E_{X} {| f (X) - {\hat{f}}_{n} (X) |}^{2}$ and $n^{- 1} \sum_{i = 1}^{n} E {(f (X_{i}) - {\hat{f}}_{n} (X_{i}))}^{2}$ are obtained.

Publié le : 1999-01-01

Zbl 0992.62039

EUDML-ID : urn:eudml:doc:219239

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     author = {Waldemar Popi\'nski},
     title = {A note on orthogonal series regression function estimators},
     journal = {Applicationes Mathematicae},
     volume = {26},
     year = {1999},
     pages = {281-291},
     zbl = {0992.62039},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p281bwm}
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Popiński, Waldemar. A note on orthogonal series regression function estimators. Applicationes Mathematicae, Tome 26 (1999) pp. 281-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv26i3p281bwm/

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