ADAPTATION IN LOG-CONCAVE DENSITY ESTIMATION

Arlene K. H. Kim

doi:10.1214/17-AOS1619

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ADAPTATION IN LOG-CONCAVE DENSITY ESTIMATION

2018-10-01

Arlene K. H. Kim *

Adityanand Guntuboyina

Richard J. Samworth

DOI：10.1214/17-AOS1619

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Abstract

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The log-concave maximum likelihood estimator of a density on the real line based on a sample of size n is known to attain the minimax optimal rate of convergence of O(n(-4/5)) with respect to, for example, squared Hellinger distance. In this paper, we show that it also enjoys attractive adaptation properties, in the sense that it achieves a faster rate of convergence when the logarithm of the true density is k-affine (i.e., made up of k-affine pieces), or close to k-affine, provided in each case that k is not too large. Our results use two different techniques: the first relies on a new Marshall's inequality for log-concave density estimation, and reveals that when the true density is close to log-linear on its support, the log-concave maximum likelihood estimator can achieve the parametric rate of convergence in total variation distance. Our second approach depends on local bracketing entropy methods, and allows us to prove a sharp oracle inequality, which implies in particular a risk bound with respect to various global loss functions, including Kullback-Leibler divergence, of O(k/n log(5/4)(en/k)) when the true density is log-concave and its logarithm is close to k-affine.

Keywords:

Adaptation

bracketing entropy

log-concavity

maximum likelihood estimation

Marshall's inequality

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