Electronic Journal of Statistics

On convex least squares estimation when the truth is linear

Yining Chen and Jon A. Wellner

Full-text: Open access

Abstract

We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process. Analogous results hold when one uses the derivative of the convex LSE to perform derivative estimation. These asymptotic results facilitate a new consistent testing procedure on the linearity against a convex alternative. Moreover, we show that the convex LSE adapts to the optimal rate at the boundary points of the region where the truth is linear, up to a log-log factor. These conclusions are valid in the context of both density estimation and regression function estimation.

Article information

Source
Electron. J. Statist. Volume 10, Number 1 (2016), 171-209.

Dates
Received: December 2014
First available in Project Euclid: 17 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1455715960

Digital Object Identifier
doi:10.1214/15-EJS1098

Mathematical Reviews number (MathSciNet)
MR3466180

Zentralblatt MATH identifier
1332.62056

Subjects
Primary: 62E20: Asymptotic distribution theory 62G07: Density estimation 62G08: Nonparametric regression 62G10: Hypothesis testing 62G20: Asymptotic properties
Secondary: 60G15: Gaussian processes

Keywords
Adaptive estimation convexity density estimation least squares regression function estimation shape constraint

Citation

Chen, Yining; Wellner, Jon A. On convex least squares estimation when the truth is linear. Electron. J. Statist. 10 (2016), no. 1, 171--209. doi:10.1214/15-EJS1098. https://projecteuclid.org/euclid.ejs/1455715960


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