On orthogonal series estimation of bounded regression functions

Waldemar Popiński. "On orthogonal series estimation of bounded regression functions." Applicationes Mathematicae 28.3 (2001): 257-270. <http://eudml.org/doc/279497>.

@article{WaldemarPopiński2001,
abstract = {The problem of nonparametric estimation of a bounded regression function $f ∈ L²([a,b]^d)$, [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_k$, k=1,2,..., is considered in the case when the observations follow the model $Y_i = f(X_i) + η_i$, i=1,...,n, where $X_i$ and $η_i$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f̂ₙ(x) = ∑_\{k=1\}^\{N(n)\} ĉ_k e_k(x)$, where the coefficients $ĉ₁,...,ĉ_\{N(n)\}$ are determined by minimizing the empirical risk $n^\{-1\} ∑_\{i=1\}^n (Y_i - ∑_\{k=1\}^\{N(n)\} c_k e_k(X_i))²$. Sufficient conditions for convergence rates of the generalization error $E_X | f(X)-f̂ₙ(X)|²$ are obtained.},
author = {Waldemar Popiński},
journal = {Applicationes Mathematicae},
keywords = {orthonormal system; least squares method; nonparametric regression; weak and strong consistency; convergence rate},
language = {eng},
number = {3},
pages = {257-270},
title = {On orthogonal series estimation of bounded regression functions},
url = {http://eudml.org/doc/279497},
volume = {28},
year = {2001},
}

TY - JOUR
AU - Waldemar Popiński
TI - On orthogonal series estimation of bounded regression functions
JO - Applicationes Mathematicae
PY - 2001
VL - 28
IS - 3
SP - 257
EP - 270
AB - The problem of nonparametric estimation of a bounded regression function $f ∈ L²([a,b]^d)$, [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions $e_k$, k=1,2,..., is considered in the case when the observations follow the model $Y_i = f(X_i) + η_i$, i=1,...,n, where $X_i$ and $η_i$ are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function $f̂ₙ(x) = ∑_{k=1}^{N(n)} ĉ_k e_k(x)$, where the coefficients $ĉ₁,...,ĉ_{N(n)}$ are determined by minimizing the empirical risk $n^{-1} ∑_{i=1}^n (Y_i - ∑_{k=1}^{N(n)} c_k e_k(X_i))²$. Sufficient conditions for convergence rates of the generalization error $E_X | f(X)-f̂ₙ(X)|²$ are obtained.
LA - eng
KW - orthonormal system; least squares method; nonparametric regression; weak and strong consistency; convergence rate
UR - http://eudml.org/doc/279497
ER -