Electronic Journal of Statistics

Contraction and uniform convergence of isotonic regression

Fan Yang and Rina Foygel Barber

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We consider the problem of isotonic regression, where the underlying signal $x$ is assumed to satisfy a monotonicity constraint, that is, $x$ lies in the cone $\{x\in \mathbb{R}^{n}:x_{1}\leq \dots\leq x_{n}\}$. We study the isotonic projection operator (projection to this cone), and find a necessary and sufficient condition characterizing all norms with respect to which this projection is contractive. This enables a simple and non-asymptotic analysis of the convergence properties of isotonic regression, yielding uniform confidence bands that adapt to the local Lipschitz properties of the signal.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 646-677.

Received: April 2018
First available in Project Euclid: 16 February 2019

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Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression 62G07: Density estimation

Isotonic regression contraction data-adaptive band convergence rates density estimation

Creative Commons Attribution 4.0 International License.


Yang, Fan; Barber, Rina Foygel. Contraction and uniform convergence of isotonic regression. Electron. J. Statist. 13 (2019), no. 1, 646--677. doi:10.1214/18-EJS1520. https://projecteuclid.org/euclid.ejs/1550286095

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