Global rates of convergence in log-concave density estimation

Abstract

The estimation of a log-concave density on R d represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size n can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order n −4/5 , when d = 1, and order n −2/(d+1) when d ≥ 2. In particular, this reveals a sense in which, when d ≥ 3, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared).Second, we show that for d ≤ 3, the Hellinger -bracketing entropy of a class of logconcave densities with small mean and covariance matrix close to the identity grows like max{ −d/2 , −(d−1) } (up to a logarithmic factor when d = 2). This enables us to prove that when d ≤ 3 the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when d = 2, 3) with respect to squared Hellinger loss.

Log-concavity and strong log-concavity: A review

“…Proposition 5.2 can also be seen as providing bounds for the variance of a log-concave variable. See Kim and Samworth [2014], section 3.2, for some further results of this type.…”

“…They show that the MLE exists and is Hellinger consistent. Doss and Wellner [2013] have obtained Hellinger rates of convergence for the maximum likelihood estimators of log-concave and s –concave densities on ℝ, while Kim and Samworth [2014] study Hellinger rates of convergence for the MLEs of log-concave densities on ℝ d . Henningsson and Astrom [2006] consider replacement of Gaussian errors by log-concave error distributions in the context of the Kalman filter.…”

A fast algorithm for univariate log‐concave density estimation

“…Dümbgen & Rufibach (2009) found that its rate of convergence with respect to the supremum norm on a compact interval is at least (log(n)=n) 1=3 . Kim & Samworth (2016) established that it achieves the minimax optimal rate with respect to squared Hellinger loss. Kim, Guntuboyina & Samworth (2017) further showed that it achieves a faster rate of convergence when the logarithm of the true density is made up of k affine pieces, or is close to being so.…”

Maximum likelihood estimation of the mixture of log-concave densities

“…Their nonparametric maximum likelihood estimators were studied by Dümbgen & Rufibach (2009), Cule et al (2010), Cule & Samworth (2010), Chen & Samworth (2013), Pal et al (2007) and Dümbgen et al (2011) (referred as [DSS 2011] thereafter). The convergence rates of these estimators for log-concave densities were studied by Doss & Wellner (2013) and Kim & Samworth (2014). Such estimators provide more generality and flexibility without any tuning parameters.…”