1 Optimal Convergence Rates in Nonparametric Regression with Fractional Time Series Errors

Consider the estimation of g(ν), the ν-th derivative of the mean function, in a fixed-design nonparametric regression model with stationary time series errors ξi. We assume that g ∈ Ck, ξi are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξi has the form f(λ) ∼ cf |λ|−α as λ → 0 with constants cf> 0 and α ∈ (−1, 1). Under regularity conditions, the optimal convergence rate of ĝ(ν) is shown to be n−rν with rν = (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 nonparametric regression with iid errors