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

摘要

Consider the estimation of g~(v), the vth derivative of the mean function, in a fixed-design non-parametric regression model with stationary time series errors ζ_i. We assume that g ∈ C~k, ζ_i are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ζ_i, has the form f(λ) ~ c_f∣λ∣~(-α) as λ → 0 with constants c_f > 0 and α ∈ (-1,1). Under regularity conditions, the optimal convergence rate of g~(v) is shown to be n~(-r) with r_v = (1 - α)(k - v)/(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.