Wavelets are particularly useful because of their natural adaptive ability to characterize data with intrinsically local properties. When the data contain outliers or come from a population with a heavy-tailed distribution, L 1-estimation should obtain a better fit. In this paper, we propose a L 1-wavelet method for nonparametric regression, and derive the asymptotic properties of the L 1-wavelet estimator, including the Bahadur representation, the rate of convergence and asymptotic normality. The rate of convergence of it is comparable with the optimal convergence rate of the nonparametric estimation in nonparametric models, and it does not require the continuously differentiable conditions of a nonparametric function.

This paper proposes wavelet-M-estimation for time-varying coefficient time series models by using a robust-type wavelet technique, which can adapt to local features of the time-varying coefficients and does not require the smoothness of the unknown time-varying coefficient. The wavelet-M-estimation has the desired asymptotic properties and can be used to estimate conditional quantile and to robustify the usual mean regression. Under mild assumptions, the Bahadur representation and the asymptotic normality of wavelet-M-estimation are established.