Title: Nonparametric Shape-restricted Regression,
Abstract: We consider the problem of nonparametric regression under shape constraints.
The main examples include isotonic regression (with respect to any partial
order), unimodal/convex regression, additive shape-restricted regression, and
constrained single index model. We review some of the theoretical properties of
the least squares estimator (LSE) in these problems, emphasizing on the
adaptive nature of the LSE. In particular, we study the behavior of the risk of
the LSE, and its pointwise limiting distribution theory, with special emphasis
to isotonic regression. We survey various methods for constructing pointwise
confidence intervals around these shape-restricted functions. We also briefly
discuss the computation of the LSE and indicate some open research problems and
future directions.