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 Σ r(j) < ∞, where r is the covariance function. Convergence rates when Σ 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|-α, the rate is n-kα/(2k+α) for 0 < α < 1 and (n-1 log n)k/(2k+1) for α = 1. These results refer to optimal convergence rates. It is shown that the optimal rates are achieved by kernel estimators. © 1990.
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