On convex least squares estimation when the truth is linear

Chen, Yining ORCID: 0000-0003-1697-1920 and Wellner, Jon A. (2016) On convex least squares estimation when the truth is linear. Electronic Journal of Statistics, 10 (1). pp. 171-209. ISSN 1935-7524

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Abstract

We prove that the convex least squares estimator (LSE) attains a n−1/2n−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.

Item Type:	Article
Official URL:	http://projecteuclid.org/ejs
Additional Information:	© 2016 The Authors © CC BY 2.5
Divisions:	Statistics
Subjects:	Q Science > QA Mathematics
Date Deposited:	14 Mar 2016 16:54
Last Modified:	17 Oct 2024 16:30
URI:	http://eprints.lse.ac.uk/id/eprint/65729

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