Local continuity of log-concave projection, with applications to estimation under model misspecification
Abstract
The log-concave projection is an operator that maps a d-dimensional distribution P to an approximating log-concave density. Prior work by D\"umbgen et al. (2011) establishes 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 H\"older-(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.