Nonparametric regression with long-range dependence

The effect of dependent errors in fixed-design, nonparameteric 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 , where r is the covariance function. Convergence rates when are also investigated. In particular, when the sample is of size n, when the mean function has k derivatives and , the rate is for and for . These results refer to optimal convergence rates. It is shown that the optimal rates are achieved by kernel estimators.