Orthogonal series regression estimation under long-range dependent errors

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. 4, pp. 457-466

Résumé

This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample $Y_i=f(X_i)+\eta _i,\ i=1,\dots,n$, where $X_i\in A\subset {\Bbb R}^d$ are independently chosen from a distribution with density $\varrho \in L^1(A)$ and $\eta _i$ are zero mean stationary errors with long-range dependence. Convergence rates of the error $n^{-1}\sum _{i=1}^n(f(X_i)-\hat f_N(X_i))^2$ for the estimator $\hat f_N(x) =\sum _{k=1}^N\hat c_ke_k(x)$, constructed using an orthonormal system $e_k,\ k=1,2,\dots,$ in $L^2(A)$, are obtained.

Détail

DOI : 10.4064/am28-4-6

Keywords: paper concerned general conditions convergence rates nonparametric orthogonal series estimators regression function estimators obtained least squares method basis observation sample x eta dots where subset bbb independently chosen distribution density varrho eta zero mean stationary errors long range dependence convergence rates error sum i hat i estimator hat sum hat constructed using orthonormal system dots obtained