ABSTRACT
Many modern problems in nonparametric statistics have focused on the estimation of smooth functions. Nonparametric density estimation considers estimating an unknown smooth density using only assumptions that address the smoothness of the unknown density and the weight of its tails. Nonparametric regression techniques focus on the estimation of an unknown smooth conditional expectation function under similar assumptions. This chapter addresses the issues of computing and assessing the accuracy of observed confidence levels for these types of problems. The regions of interest in these problems often consider the density or regression function itself as the parameter of interest. As such, the parameter space becomes a space of functions, and the regions of interest often correspond to subsets of the function space that contain functions with particular properties. For example, in nonparametric density estimation interest may lie in assessing the amount of confidence there is, based on the observed data, that the unknown population density has one, two or three modes. In nonparametric regression problems it may be of interest to assess the amount confidence there is that the regression function lies within a certain region.
