Adaptive Confidence Bands for Nonparametric Regression Functions

Abstract

This article proposes a new formulation for the construction of adaptive confidence bands (CBs) in nonparametric function estimation problems. CBs, which have size that adapts to the smoothness of the function while guaranteeing that both the relative excess mass of the function lying outside the band and the measure of the set of points where the function lies outside the band are small. It is shown that the bands adapt over a maximum range of Lipschitz classes. The adaptive CB can be easily implemented in standard statistical software with wavelet support. We investigate the numerical performance of the procedure using both simulated and real datasets. The numerical results agree well with the theoretical analysis. The procedure can be easily modified and used for other nonparametric function estimation models. Supplementary materials for this article are available online.