Electronic Journal of Statistics

Contraction and uniform convergence of isotonic regression

Abstract

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

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

Dates
First available in Project Euclid: 16 February 2019

https://projecteuclid.org/euclid.ejs/1550286095

Digital Object Identifier
doi:10.1214/18-EJS1520

Mathematical Reviews number (MathSciNet)
MR3914177

Zentralblatt MATH identifier
07038000

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

Citation

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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