The estimation of a log-concave density on $\mathbb{R}^d$ represents acentral problem in the area of nonparametric inference under shape constraints.In this paper, we study the performance of log-concave density estimators withrespect to global loss functions, and adopt a minimax approach. We first showthat no statistical procedure based on a sample of size $n$ can estimate alog-concave density with respect to the squared Hellinger loss function withsupremum 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 morechallenging than the estimation of a density with two bounded derivatives (aproblem to which it has been compared). Second, we show that for $d \leq 3$,the Hellinger $\epsilon$-bracketing entropy of a class of log-concave densitieswith small mean and covariance matrix close to the identity grows like$\max\{\epsilon^{-d/2},\epsilon^{-(d-1)}\}$ (up to a logarithmic factor when$d=2$). This enables us to prove that when $d \leq 3$ the log-concave maximumlikelihood estimator achieves the minimax optimal rate (up to logarithmicfactors when $d = 2,3$) with respect to squared Hellinger loss.
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机译：$ \ mathbb {R} ^ d $上对数凹面密度的估计代表了形状约束下非参数推断领域的一个中心问题。本文研究了整体损失函数对数凹面密度估计器的性能，并采用minimax方法。我们首先证明，当$ d = 1时，没有基于大小为$ n $的样本的统计程序可以估计平方平方的相对于Hellinger损失函数的凹模密度，且最高风险小于$ n ^ {-4/5} $。 $，并在$ d \ geq 2 $时订购$ n ^ {-2 /（d + 1）} $。特别是，这揭示了一种感觉，当$ d \ geq 3 $时，对数凹面密度估计比具有两个有界导数的密度估计（具有比较性的方法）更具挑战性。其次，我们证明对于$ d \ leq 3 $，一类对数凹面密度的Hellinger $ \ epsilon $包围式熵具有较小的均值和接近方差的协方差矩阵，其增长为$ \ max \ {\ epsilon ^ { -d / 2}，\ epsilon ^ {-（d-1）} \} $（当$ d = 2 $时为对数因子）。这使我们能够证明，当$ d \ leq 3 $时，相对于平方的Hellinger损失，对数凹面最大似然估计值达到了最小最大最优速率（当$ d = 2,3 $时，达到对数因子）。
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