Global rates of convergence in log-concave density estimation

Understand

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 $d \geq 2$.
• In particular, this reveals a sense in which, when $d \geq 3$, 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).