%0 Journal Article
%T A Note on Wavelet Estimation of the Derivatives of a Regression Function in a Random Design Setting
%A Christophe Chesneau
%J International Journal of Mathematics and Mathematical Sciences
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/195765
%X We investigate the estimation of the derivatives of a regression function in the nonparametric regression model with random design. New wavelet estimators are developed. Their performances are evaluated via the mean integrated squared error. Fast rates of convergence are obtained for a wide class of unknown functions. 1. Introduction We consider the nonparametric regression model with random design described as follows. Let be random variables defined on a probability space , where are i.i.d. random variables such that and , are i.i.d. random variables with common density , and is an unknown regression function. It is assumed that and are independent for any . We aim to estimate , that is, the th derivative of , for any integer , from . In the literature, various estimation methods have been proposed and studied. The main ones are the kernel methods (see, e.g., [1¨C5]), the smoothing splines, and local polynomial methods (see, e.g., [6¨C9]). The object of this note is to introduce new efficient estimators based on wavelet methods. Contrary to the others, they have the benefit of enjoying local adaptivity against discontinuities thanks to the use of a multiresolution analysis. Reviews on wavelet methods can be found in, for example, Antoniadis [10], H£¿rdle et al. [11], and Vidakovic [12]. To the best of our knowledge, only Cai [13] and Petsa and Sapatinas [14] have proposed wavelet estimators for from (1) but defined with a deterministic equidistant design; that is, . The consideration of a random design complicates significantly the problem and no wavelet estimators exist in this case. This motivates our study. In the first part, assuming that is known, we propose two wavelet estimators: the first one is linear nonadaptive and the second one nonlinear adaptive. Both use the approach of Prakasa Rao [15] initially developed in the context of the density estimation problem. Then we determine their rates of convergence by considering the mean integrated squared error (MISE) and assuming that belongs to Besov balls. In a second part, we develop a linear wavelet estimator in the case where is unknown. It is derived from the one introduced by Pensky and Vidakovic [16] considering the estimation of from (1). We evaluate its rate of convergence again under the MISE over Besov balls. The obtained rates of convergence are similar to those attained by wavelet estimators for the derivatives of a density (see, e.g., [15, 17, 18]). The organization of this note is as follows. The next section describes some basics on wavelets and Besov balls. Our estimators and their
%U http://www.hindawi.com/journals/ijmms/2014/195765/