2016/09/03 by Arlene K. H. Kim, Adityanand Guntuboyina, Kim, Arlene K. H. +3 · 1 citation
Computer Science · Mathematics · #62G05 #62G07 #FOS: Mathematics #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1609.00861
openalex publication_date 2016/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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), provided 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 that the rate of convergence with respect to various global loss functions, including Kullback--Leibler divergence, is O((k)/(n)log5/4 n) when the true density is log-concave and its logarithm is close to k-affine.