Abstract
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.
Highlights
Consider the problem of estimating the underlying regression function from a set of noisy data
We aim to study the asymptotic properties on L1-wavelet estimator for the nonparametric model (1.1)
The convergence rate and asymptotic normality of wavelet estimators were considered by [22, 23] provided asymptotic bias and variance of wavelet estimator for regression function under a mixing stochastic process
Summary
1 Introduction Consider the problem of estimating the underlying regression function from a set of noisy data. Most methods developed so far are based on the mean regression function by using a least-squares estimation (L2). We aim to study the asymptotic properties on L1-wavelet estimator for the nonparametric model (1.1).
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