Global rates of convergence in log-concave density estimation

By Arlene K. H. Kim, Richard J. Samworth. 2014

107 citations recorded by the source corpus.

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). Second, we show that for $d \leq 3$, the Hellinger $\epsilon$-bracketing entropy of a class of log-concave densities with 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 maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when $d = 2,3$) with respect to squared Hellinger loss.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/61777e64b9c2653c672e8833469e5319d2fd9799

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Citations in the corpus

Generalized additive and index models with shape constraints (2014, methodology): “Recent work by Kim and Samworth (2014) has shown that shape-restricted inference without further assumptions can lead to slow rates of convergence in higher dimensions.”
Recent Progress in Log-Concave Density Estimation (2017, background): “We mention, however, that lower bounds on the bracketing entropy obtained in Kim and Samworth (2016) strongly suggest, but do not prove, that the log-concave MLE will be rate-suboptimal when d ≥ 4.”
Log-Concavity and Strong Log-Concavity: a review (2014, background): “Indeed, it ensures the existence of super-Gaussian marginals in every directions (see Klartag and Milman (2012) for a definition of super-Gaussianity of marginals).”
Faithful Variable Screening for High-Dimensional Convex Regression (2014, methodology): “We also include an extra boundedness constraint to use new bracketing number results (Kim and Samworth, 2014).”