Open Access
August 2019 On the risk of convex-constrained least squares estimators under misspecification
Billy Fang, Adityanand Guntuboyina
Bernoulli 25(3): 2206-2244 (August 2019). DOI: 10.3150/18-BEJ1051


We consider the problem of estimating the mean of a noisy vector. When the mean lies in a convex constraint set, the least squares projection of the random vector onto the set is a natural estimator. Properties of the risk of this estimator, such as its asymptotic behavior as the noise tends to zero, have been well studied. We instead study the behavior of this estimator under misspecification, that is, without the assumption that the mean lies in the constraint set. For appropriately defined notions of risk in the misspecified setting, we prove a generalization of a low noise characterization of the risk due to [Found. Comput. Math. 16 (2016) 965–1029] in the case of a polyhedral constraint set. An interesting consequence of our results is that the risk can be much smaller in the misspecified setting than in the well-specified setting. We also discuss consequences of our result for isotonic regression.


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Billy Fang. Adityanand Guntuboyina. "On the risk of convex-constrained least squares estimators under misspecification." Bernoulli 25 (3) 2206 - 2244, August 2019.


Received: 1 June 2017; Revised: 1 February 2018; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066255
MathSciNet: MR3961246
Digital Object Identifier: 10.3150/18-BEJ1051

Keywords: convex constraint , isotonic regression , least squares , misspecification , statistical dimension

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
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