NONPARAMETRIC REGRESSION WITH LONG-RANGE DEPENDENCE
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The effect of dependent errors in fixed-design, nonparametric regression is investigated. It is shown that convergence rates for a regression mean estimator under the assumption of independent errors are maintained in the presence of stationary dependent errors, if and only if r(j) < , where r is the covariance function. Convergence rates when r(j) = are also investigated. In particular, when the sample is of size n, when the mean function has k derivatives and r(j) C|j|-, the rate is n-k/(2k+) for 0 < < 1 and (n-1 log n)k/(2k+1) for = 1. These results refer to optimal convergence rates. It is shown that the optimal rates are achieved by kernel estimators. 1990.