Statistical Science

Nonparametric Shape-Restricted Regression

Adityanand Guntuboyina and Bodhisattva Sen

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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.

Article information

Source
Statist. Sci., Volume 33, Number 4 (2018), 568-594.

Dates
First available in Project Euclid: 29 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ss/1543482059

Digital Object Identifier
doi:10.1214/18-STS665

Mathematical Reviews number (MathSciNet)
MR3881209

Zentralblatt MATH identifier
07032830

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

Citation

Guntuboyina, Adityanand; Sen, Bodhisattva. Nonparametric Shape-Restricted Regression. Statist. Sci. 33 (2018), no. 4, 568--594. doi:10.1214/18-STS665. https://projecteuclid.org/euclid.ss/1543482059


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Supplemental materials

  • Supplement to “Nonparametric Shape-Restricted Regression”. The supplement contains some of the detailed proofs of results in the paper.