28 Sep 2017

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, e.g., 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 log5/4 (en/k) when the true density is log-concave and its logarithm is close to k-affine.