On nonparametric estimation in nonlinear AR(1)-models

We estimate the mean function and the conditional variance (the volatility function) of a nonlinear first-order autoregressive model nonparametrically. Minimax rates of convergence are established over a scale of Besov bodies Bspq and a range of global Lp' error measurements, for 1[less-than-or-equals, slant]p'<[infinity]. We propose an estimating procedure based on a martingale regression approximation scheme. This enables us to implement wavelet thresholding and obtain adaptation results with respect to an unknown degree of smoothness.