On the adaptive wavelet estimation of a multidimensional regression function under $\alpha$-mixing dependence: Beyond the standard assumptions on the noise

Commentationes mathematicae Universitatis Carolinae

Christophe Chesneau¹ 

关键词: nonparametric regression;    $\alpha$-mixing dependence;    adaptive estimation;    wavelet methods;    rates of convergence;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of an $\alpha$-mixing process $(Y,X)$, where $Y=f(X)+\xi$, $X$ represents the design and $\xi$ the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of $f$ in its construction) or it is supposed that $\xi$ is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework. Under no boundedness assumption on $\xi$ and no a priori knowledge on its distribution, we construct adaptive term-by-term thresholding wavelet estimators attaining ``sharp'' rates of convergence under the mean integrated squared error over a wide class of functions $f$.

【 授权许可】

CC BY