On the adaptive wavelet estimation of a multidimensional regression function under \(\alpha \)-mixing dependence: beyond the standard assumptions on the noise. (English) Zbl 1313.62058
Let \(\{(Y_t,X_t)\}_{t\in \mathbb {Z}}\) be a strictly stationary process with \(Y_t=f(X_t)+\xi _t\), where \(f:[0,1]^d\to \mathbb {R}\) is the unknown regression function and \(\xi _t\) is the noise. In regression analysis, one wants to estimate the regression function from a data set \(\{(Y_i,X_i)\}_{i=1}^n\) drawn from \(\{(Y_t,X_t)\}_{t\in \mathbb {Z}}\). In many applications, it is necessary to consider that the data set \(\{(Y_i,X_i)\}_{i=1}^n\) is dependent. Many kinds of mixing dependences as the \(\alpha \)-mixing dependence and the \(\beta \)-mixing dependence are considered. In this paper, when the data set \(\{(Y_i,X_i)\}_{i=1}^n\) is the \(\alpha \)-mixing dependence, \(\xi _1\) is not necessarily bounded and its distribution is not necessarily known, and the regression function is included in Besov balls, the fast rate of convergence of the adaptive wavelet estimators are given, i.e., the construction of the estimators does not depend on the unknown distribution.
Reviewer: Takanori Ayano (Osaka)
MSC:
62G08 | Nonparametric regression and quantile regression |
42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
60G10 | Stationary stochastic processes |
62G20 | Asymptotic properties of nonparametric inference |