On orthogonal series estimation of bounded regression functions
Volume 28 / 2001
Abstract
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.
