On orthogonal series estimation of bounded regression functions

Waldemar Popi/nski¹

¹ Department of Survey Design Central Statistical Office Al. Niepodleg/lo/sci 208 00-925 Warszawa, Poland

Applicationes Mathematicae, Tome 28 (2001) no. 3, pp. 257-270

Résumé

The problem of nonparametric estimation of a bounded regression function $f\in L^2([a,b]^d),\ [a,b]\subset {\Bbb R},\ d\geq 1$, using an orthonormal system of functions $e_k,\ k=1,2,\mathinner {\ldotp \ldotp \ldotp },$ is considered in the case when the observations follow the model $Y_i=f(X_i)+\eta _i,\ i=1,\mathinner {\ldotp \ldotp \ldotp },n$, where $X_i$ and $\eta _i$ are i.i.d. copies of independent random variables $X$ and $\eta $, respectively, the distribution of $X$ has density $\varrho $, and $\eta $ has mean zero and finite variance. The estimators are constructed by proper truncation of the function $\hat f_n(x) = \sum _{k=1}^{N(n)}\hat c_ke_k(x)$, where the coefficients $\hat c_1,\mathinner {\ldotp \ldotp \ldotp },\hat c_{N(n)}$ are determined by minimizing the empirical risk $n^{-1}\sum _{i=1}^n(Y_i-\sum _{k=1}^{N(n)}c_ke_k(X_i))^2$. Sufficient conditions for convergence rates of the generalization error $E_X| f(X)-\hat f_n(X)|^2$ are obtained.

Détail

DOI : 10.4064/am28-3-2

Keywords: problem nonparametric estimation bounded regression function subset bbb geq using orthonormal system functions mathinner ldotp ldotp ldotp considered observations follow model x eta mathinner ldotp ldotp ldotp where eta copies independent random variables eta respectively distribution has density varrho eta has mean zero finite variance estimators constructed proper truncation function hat sum hat where coefficients hat mathinner ldotp ldotp ldotp hat determined minimizing empirical risk sum i sum sufficient conditions convergence rates generalization error hat obtained