Global rates of convergence in log-concave density estimation

Abstract

The estimation of a log-concave density on $\mathbb{R}^{d}$ 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 $d\geq2$. In particular, this reveals a sense in which, when $d\geq3$, 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 $d\leq3$, the Hellinger $\varepsilon$-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like $\max\{\varepsilon^{-d/2},\varepsilon^{-(d-1)}\}$ (up to a logarithmic factor when $d=2$). This enables us to prove that when $d\leq3$ 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.