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November 2018 Nonparametric Shape-Restricted Regression
Adityanand Guntuboyina, Bodhisattva Sen
Statist. Sci. 33(4): 568-594 (November 2018). DOI: 10.1214/18-STS665


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.


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Adityanand Guntuboyina. Bodhisattva Sen. "Nonparametric Shape-Restricted Regression." Statist. Sci. 33 (4) 568 - 594, November 2018.


Published: November 2018
First available in Project Euclid: 29 November 2018

zbMATH: 07032830
MathSciNet: MR3881209
Digital Object Identifier: 10.1214/18-STS665

Keywords: adaptive risk bounds , bootstrap , Chernoff’s distribution , convex regression , isotonic regression , likelihood ratio test , monotone function , order preserving function estimation , projection on a closed convex set , tangent cone

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.33 • No. 4 • November 2018
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