On orthogonal series estimation of bounded regression functions

Waldemar Popiński

Applicationes Mathematicae, Tome 28 (2001), p. 257-270 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

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Publié le : 2001-01-01

Zbl 1008.62038

EUDML-ID : urn:eudml:doc:279497