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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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Identification Number: 10.1214/15-EJS1098


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
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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: 01 Jun 2024 03:03

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