Nonparametric Shape-Restricted Regression
Copyright policy )Nonparametric Shape-Restricted Regression
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.
This is a survey paper
- University of California, Berkeley United States
- King’s University United States
FOS: Computer and information sciences, projection on a closed convex set, isotonic regression, Mathematics - Statistics Theory, Machine Learning (stat.ML), Statistics Theory (math.ST), Chernoff’s distribution, Statistics - Machine Learning, Asymptotic properties of nonparametric inference, convex regression, FOS: Mathematics, Nonparametric regression and quantile regression, Adaptive risk bounds, bootstrap, tangent cone, Chernoff's distribution, order preserving function estimation, likelihood ratio test, monotone function, adaptive risk bounds
FOS: Computer and information sciences, projection on a closed convex set, isotonic regression, Mathematics - Statistics Theory, Machine Learning (stat.ML), Statistics Theory (math.ST), Chernoff’s distribution, Statistics - Machine Learning, Asymptotic properties of nonparametric inference, convex regression, FOS: Mathematics, Nonparametric regression and quantile regression, Adaptive risk bounds, bootstrap, tangent cone, Chernoff's distribution, order preserving function estimation, likelihood ratio test, monotone function, adaptive risk bounds
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