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Abstract

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.