Orthogonal series estimation of band-limited regression functions
Volume 41 / 2014
Abstract
The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions $s_k$, $k=0,\pm 1,\ldots ,$ for the observation model $y_j = f(u_j) + \eta _j $, $j=1,\ldots ,n$, is considered, where $f\in L^2(\mathbb {R})\cap BL(\varOmega )$ for $\varOmega >0$ is a band-limited function, $u_j$ are independent random variables uniformly distributed in the observation interval $[-T,T]$, $\eta _j$ are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions for convergence and convergence rates of the integrated mean-square error $E\| f-\hat f_n\| ^2$ and the pointwise mean-square error $E(f(x)-\hat f_n(x))^2$ of the estimator $\hat f_n(x) = \sum _{k=-N(n)}^{N(n)}\hat c_ks_k(x)$ with coefficients $\hat c_k$, $k=-N(n),\ldots ,N(n)$, obtained by the Monte Carlo method are studied.
