Orthogonal series estimation of band-limited regression functions

Waldemar Popiński¹

¹ Space Research Centre Polish Academy of Sciences Bartycka 18a 00-716 Warszawa, Poland

Applicationes Mathematicae, Tome 41 (2014) no. 1, pp. 51-65

Résumé

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.

Détail

DOI : 10.4064/am41-1-5

Keywords: problem nonparametric function fitting using complete orthogonal system whittaker cardinal functions ldots observation model eta ldots considered where mathbb cap varomega varomega band limited function independent random variables uniformly distributed observation interval t eta uncorrelated correlated random variables zero mean value finite variance independent observation points conditions convergence convergence rates integrated mean square error f hat pointwise mean square error hat estimator hat sum n hat coefficients hat n ldots obtained monte carlo method studied