Asymptotics for $L_{1}$-wavelet method for nonparametric regression

Abstract

AbstractWavelets 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, $L_{1}$ L 1 -estimation should obtain a better fit. In this paper, we propose a $L_{1}$ L 1 -wavelet method for nonparametric regression, and derive the asymptotic properties of the $L_{1}$ 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.