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November 2021 Local continuity of log-concave projection, with applications to estimation under model misspecification
Rina Foygel Barber, Richard J. Samworth
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Bernoulli 27(4): 2437-2472 (November 2021). DOI: 10.3150/20-BEJ1316


The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. It is known that, with suitable metrics on the underlying spaces, this projection is continuous, but not uniformly continuous. In this work, we prove a local uniform continuity result for log-concave projection – in particular, establishing that this map is locally Hölder-(1/4) continuous. A matching lower bound verifies that this exponent cannot be improved. We also examine the implications of this continuity result for the empirical setting – given a sample drawn from a distribution P, we bound the squared Hellinger distance between the log-concave projection of the empirical distribution of the sample, and the log-concave projection of P. In particular, this yields interesting statistical results for the misspecified setting, where P is not itself log-concave.

Funding Statement

R.F.B. was supported by the National Science Foundation via grant DMS-1654076 and by an Alfred P. Sloan fellowship. R.J.S. was supported by EPSRC grants EP/P031447/1 and EP/N031938/1.


The authors thank the anonymous reviewers and Oliver Feng for helpful comments.


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Rina Foygel Barber. Richard J. Samworth. "Local continuity of log-concave projection, with applications to estimation under model misspecification." Bernoulli 27 (4) 2437 - 2472, November 2021.


Received: 1 July 2020; Revised: 1 December 2020; Published: November 2021
First available in Project Euclid: 24 August 2021

Digital Object Identifier: 10.3150/20-BEJ1316

Keywords: Hellinger distance , Hölder continuity , Log-concavity , maximum likelihood estimation , Wasserstein distance

Rights: Copyright © 2021 ISI/BS


Vol.27 • No. 4 • November 2021
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