Statistica Sinica 20 (2010), 1771-1787

ADAPTIVE GLOBAL CONFIDENCE BAND FOR

NONPARAMETRIC REGRESSION:

AN EMPIRICAL LIKELIHOOD METHOD

Li-Xing Zhu$^1$, Lu Lin$^2$ and Qiang Chen$^{2,3}$

$^1$Hong Kong Baptist University, $^2$Shandong University and

$^3$Qilu Securities Co., Ltd

Abstract: In this paper, we construct adaptive global confidence bands for nonparametric regression functions by empirical likelihood (EL). First, we show that the size of the classical EL-based confidence region is not adaptive to the submodels of the function in rate-optimal way, that is, it is not model-adaptive. In contrast, the existing model-adaptive methods are not data-adaptive, that is, the shapes of the resulting confidence regions are not determined by data. Thus, we propose an EL-based method to construct model-data-adaptive global confidence bands for nonparametric regression models with some constraints. The key remark is that the size (radius) of the confidence region is not determined by the (asymptotic) distribution but by a $U$-statistic that is highly related to the smoothness of the submodels. The newly proposed confidence region has the model-data-adaptive property: the size adapts to the submodels in a rate-optimal way and its shape is determined by the data. Implementation issue is investigated, and simulations are carried out for illustration.