Multivariate log-concave distributions as a nearly parametric model

Schuhmacher, Dominic; Huesler, Andre; Duembgen, Lutz

doi:10.1524/stnd.2011.1073

Cited by 17 publications

(25 citation statements)

References 25 publications

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“…Similar results were proved in Theorem 2.1 and Proposition 2.2 of Schuhmacher, Hüsler and Dümbgen (2011) under the stronger assumption that ν has a log-concave RadonNikodym derivative with respect to µ d .…”

Section: Convergence Of Log-concave Densitiessupporting

confidence: 80%

Global rates of convergence in log-concave density estimation

Kim¹,

Samworth²

2016

Ann. Statist.

109

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The estimation of a log-concave density on 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 ≥ 2. In particular, this reveals a sense in which, when 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 d ≤ 3, the Hellinger -bracketing entropy of a class of logconcave densities with small mean and covariance matrix close to the identity grows like max{ −d/2 , −(d−1) } (up to a logarithmic factor when d = 2). This enables us to prove that when d ≤ 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.

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Section: Convergence Of Log-concave Densitiessupporting

confidence: 80%

Global rates of convergence in log-concave density estimation

Kim¹,

Samworth²

2016

Ann. Statist.

109

View full text Add to dashboard Cite

show abstract

“…Walther (2009) provides a nice recent review of inference and modelling with log‐concave densities. Other recent related work includes Seregin and Wellner (2010), Schuhmacher et al. (2009), Schuhmacher and Dümbgen (2010) and Koenker and Mizera (2010).…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation of a Multi-Dimensional Log-Concave Density

Cule

Samworth

Stewart

2010

Journal of the Royal Statistical Society Series B: Statistical Methodology

180

241

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Summary. Let X 1 , ..., X n be independent and identically distributed random vectors with a (Lebesgue) density f. We first prove that, with probability 1, there is a unique log-concave maximum likelihood estimatorf n of f. The use of this estimator is attractive because, unlike kernel density estimation, the method is fully automatic, with no smoothing parameters to choose. Although the existence proof is non-constructive, we can reformulate the issue of computingf n in terms of a non-differentiable convex optimization problem, and thus combine techniques of computational geometry with Shor's r -algorithm to produce a sequence that converges tof n . An R version of the algorithm is available in the package LogConcDEAD-log-concave density estimation in arbitrary dimensions. We demonstrate that the estimator has attractive theoretical properties both when the true density is log-concave and when this model is misspecified. For the moderate or large sample sizes in our simulations,f n is shown to have smaller mean integrated squared error compared with kernel-based methods, even when we allow the use of a theoretical, optimal fixed bandwidth for the kernel estimator that would not be available in practice. We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation-maximization algorithm to fit finite mixtures of log-concave densities.

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“…Further study of the class of log-concave densities from this perspective has been undertaken by Schuhmacher, Hüsler and Duembgen (2009).…”

Section: Introductionmentioning

confidence: 99%

Nonparametric estimation of multivariate convex-transformed densities

Seregin¹,

Wellner²

2010

Ann. Statist.

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We study estimation of multivariate densities p of the form p(x) = h(g(x)) for x ∈ ℝd and for a fixed monotone function h and an unknown convex function g. The canonical example is h(y) = e−y for y ∈ ℝ; in this case, the resulting class of densities P(e−y)={p=exp(−g):gis convex}is well known as the class of log-concave densities. Other functions h allow for classes of densities with heavier tails than the log-concave class. We first investigate when the maximum likelihood estimator p̂ exists for the class P(h) for various choices of monotone transformations h, including decreasing and increasing functions h. The resulting models for increasing transformations h extend the classes of log-convex densities studied previously in the econometrics literature, corresponding to h(y) = exp(y). We then establish consistency of the maximum likelihood estimator for fairly general functions h, including the log-concave class P(e−y) and many others. In a final section, we provide asymptotic minimax lower bounds for the estimation of p and its vector of derivatives at a fixed point x0 under natural smoothness hypotheses on h and g. The proofs rely heavily on results from convex analysis.

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Multivariate log-concave distributions as a nearly parametric model

Cited by 17 publications

References 25 publications

Global rates of convergence in log-concave density estimation

Global rates of convergence in log-concave density estimation

Maximum Likelihood Estimation of a Multi-Dimensional Log-Concave Density

Nonparametric estimation of multivariate convex-transformed densities

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