Nonparametric Shape-restricted Regression
This is an incremental review that surveys existing methods and open problems in shape-restricted regression for statistical modeling applications.
The paper reviews theoretical properties of the least squares estimator for nonparametric regression under shape constraints, such as isotonic and convex regression, focusing on risk behavior and pointwise limiting distribution theory.
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.