Journal of Time Series Analysis

Optimal convergence rates in non‐parametric regression with fractional time series errors

Journal Article

Consider the estimation of g(ν), the νth derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξi. We assume that inline image, ξi are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξi has the form inline image as λ → 0 with constants cf > 0 and α  ∈  (−1,1). Under regularity conditions, the optimal convergence rate of inline image is shown to be inline image with r = (1 − α)(k − ν)/(2k+1 − α). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.

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