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On convex least squares estimation when the truth is linear

Chen, Yining 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: 20 Aug 2021 02:09

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