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Global Rates of Convergence in Log-Concave Density Estimation

dc.creatorKim, Arlene KH
dc.creatorSamworth, Richard John
dc.date.accessioned2016-05-24
dc.date.accessioned2018-11-24T23:26:47Z
dc.date.available2016-06-23T11:17:00Z
dc.date.available2018-11-24T23:26:47Z
dc.date.issued2016
dc.identifierhttps://www.repository.cam.ac.uk/handle/1810/256453
dc.identifier.urihttp://repository.aust.edu.ng/xmlui/handle/123456789/3886
dc.description.abstractThe estimation of a log-concave density on $\Bbb R$$^d$ represents a central problem in the area of nonparametric inference under shape constraints. In this paper, we study the performance of log-concave density estimators with respect to global loss functions, and adopt a minimax approach. We first show that no statistical procedure based on a sample of size $\textit{n}$ can estimate a log-concave density with respect to the squared Hellinger loss function with supremum risk smaller than order $\textit{n}$$^{−4/5}$ , when $\textit{d}$ = 1, and order $\textit{n}$$^{−2/(d+1)}$ when $\textit{d}$ ≥ 2. In particular, this reveals a sense in which, when $\textit{d}$ ≥ 3, log-concave density estimation is fundamentally more challenging than the estimation of a density with two bounded derivatives (a problem to which it has been compared). Second, we show that for $\textit{d}$ ≤ 3, the Hellinger $\epsilon$-bracketing entropy of a class of log-concave densities with small mean and covariance matrix close to the identity grows like max{$\epsilon$$^{−d/2}$ , $\epsilon$$^{−(d−1)}$} (up to a logarithmic factor when $\textit{d}$ = 2). This enables us to prove that when $\textit{d}$ ≤ 3 the log-concave maximum likelihood estimator achieves the minimax optimal rate (up to logarithmic factors when $\textit{d}$ = 2, 3) with respect to squared Hellinger loss.
dc.languageen
dc.publisherInstitute of Mathematical Statistics
dc.publisherAnnals of Statistics
dc.subjectbracketing entropy
dc.subjectdensity estimation
dc.subjectglobal loss function
dc.subjectlog-concavity
dc.subjectmaximum likelihood estimation
dc.titleGlobal Rates of Convergence in Log-Concave Density Estimation
dc.typeArticle


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