Optimal Convergence Rates in Non‐Parametric Regression with Fractional Time Series Errors
10 Pages Posted: 23 Dec 2012
Date Written: January 2013
Consider the estimation of g(v), the νth derivative of the mean function, in a fixed‐design non‐parametric 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 ^g(v) is shown to be n-rv with rv=(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.
Keywords: Optimal rate of convergence, non‐parametric regression, long memory, antipersistence
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