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

Christophe Chesneau (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We investigate the estimation of a multidimensional regression function $f$ from $n$ observations of a $\alpha$-mixing process $(Y,X)$ with $Y=f(X)+\xi$, where $X$ represents the design and $\xi$ the noise. We focus our attention 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 (excluding the Gaussian case). In this study we go far beyond this classical framework. Under no boundedness assumption on $\xi$, we construct adaptive term-by-term thresholding wavelet estimators enjoying powerful mean integrated squared error (MISE) properties. More precisely, we prove that they achieve "sharp" rates of convergence under the MISE over a wide class of functions $f$.

Mots clés: Nonparametric regression; $\alpha$-mixing dependence; Adaptive estimation; Wavelet methods; Rates of convergence

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Christophe Chesneau. On the adaptive wavelet estimation of a multidimensional regression function under $\alpha$-mixing dependence: Beyond the boundedness assumption on the noise. 2013. ⟨hal-00727150v3⟩