%0 Journal Article
%T A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator under -Mixing Dependence
%A Christophe Chesneau
%J Journal of Probability and Statistics
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/403764
%X We consider the estimation of an unknown function for weakly dependent data ( -mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design. 1. Introduction A general nonparametric problem is adopted: we aim to estimate an unknown function via random variables from a strictly stationary stochastic process . We suppose that has a weak dependence structure; the -mixing case is considered. This kind of dependence naturally appears in numerous models as Markov chains, GARCH-type models, and discretely observed diffusions (see, e.g., [1¨C3]). The problems where is the density of or a regression function have received a lot of attention. A partial list of related works includes Robinson [4], Roussas [5, 6], Truong and Stone [7], Tran [8], Masry [9, 10], Masry and Fan [11], Bosq [12], and Liebscher [13]. For an efficient estimation of , many methods can be considered. The most popular of them are based on kernels, splines and wavelets. In this note we deal with wavelet methods that have been introduced in i.i.d.setting by Donoho and Johnstone [14, 15] and Donoho et al. [16, 17]. These methods enjoy remarkable local adaptivity against discontinuities and spatially varying degree of oscillations. Complete reviews and discussions on wavelets in statistics can be found in, for example, Antoniadis [18] and H£¿rdle et al. [19]. In the context of -mixing dependence, various wavelet methods have been elaborated for a wide variety of nonparametric problems. Recent developments can be found in, for example, Leblanc [20], Tribouley and Viennet [21], Masry [22], Patil and Truong [23], Doosti et al. [24], Doosti and Niroumand [25], Doosti et al. [26], Cai and Liang [27], Niu and Liang [28], Benatia and Yahia [29], Chesneau [30¨C32], Chaubey and Shirazi [33], and Abbaszadeh and Emadi [34]. In the general dependent setting described above, we provide a theoretical contribution to the performance of a wavelet estimator based on a hard thresholding. This nonlinear wavelet procedure has the features to be fully adaptive and efficient over a large class of functions (see, e.g., [14¨C17, 35]). Following the
%U http://www.hindawi.com/journals/jps/2014/403764/