Statistica Sinica 19 (2009), 1371-1387

SMOOTHING NON-EQUISPACED HEAVY NOISY DATA

WITH WAVELETS

Anestis Antoniadis, Irène Gijbels and Jean-Michel Poggi

Université Joseph Fourier, Katholieke Universiteit Leuven

and Université Paris-Sud

Abstract: We consider a nonparametric noisy data model $Y_k=f(x_k)+\epsilon_k$, $k=1, \ldots, n$, where the unknown signal $f: \ [0,1] \rightarrow{\mathbb{R}}$ is assumed to belong to a wide range of function classes, including discontinuous functions, and the $\epsilon_k's$ are independent identically distributed noises with zero median. The distribution of the noise is assumed to be unknown and to satisfy some weak conditions. Possible noise distributions may have heavy tails, so that, for example, no moments of the noise exist. The design points are assumed to be deterministic points, not necessarily equispaced within the interval $[0, 1]$. Since the functions can be nonsmooth and the noise may have heavy tails, traditional estimation methods (for example, kernel methods) cannot be applied directly in this situation. As in Brown, Cai, and Zhou (2008), our approach first uses local medians to construct certain variables $Z_k$ structured as a Gaussian nonparametric regression but, unlike in this paper, the resulting data being not equispaced, we apply a wavelet block penalizing procedure adapted to non-equidistant designs to construct an estimator of the regression function. Under mild assumptions on the design it is shown that our estimator simultaneously attains the optimal rate of convergence over a wide range of Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The performance of our procedures is evaluated on simulated data sets covering a broad variety of settings and on some data examples, and are compared with other proposals made in the literature for treating similar problems.

Key words and phrases: Median, non equispaced design, penalization, robust regression, wavelets.