Orthogonal series regression estimation under long-range dependent errors

Volume 28 / 2001

Waldemar Popi/nski Applicationes Mathematicae 28 (2001), 457-466 MSC: 62G08, 62G20. DOI: 10.4064/am28-4-6

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.

Authors

  • Waldemar Popi/nskiDepartment of Survey Design
    Central Statistical Office
    Al. Niepodleg/lo/sci 208
    00-925 Warszawa, Poland
    e-mail

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