Global rates of convergence of the MLEs of log-concave and $s$-concave densities

Doss, Charles R.; Wellner, Jon A.

doi:10.1214/15-aos1394

Cited by 56 publications

(63 citation statements)

References 35 publications

(97 reference statements)

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“…This is a stark contrast with the univariate setting studied by Doss and Wellner (2015), where a similar general strategy was introduced, but where only O( −2 ) brackets are needed for the domains.…”

Section: Integrable Envelopes For Classes Of Log-concave Densitiesmentioning

confidence: 99%

“…They proved that the log-concave maximum likelihood estimatorf n based on a sample of size n converges in these metrics to f 0 at rate O p (ρ −β/(2β+1) n ), where ρ n := n/ log n; thusf n attains the same rates in the stated regimes as other adaptive nonparametric estimators that do not satisfy the shape constraint. Very recently, Doss and Wellner (2015) introduced a new bracketing argument to obtain a rate of convergence of O p (n −4/5 ) in squared Hellinger distance in the case d = 1, again for a fixed true log-concave density f 0 .…”

Section: Introductionmentioning

confidence: 99%

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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: Integrable Envelopes For Classes Of Log-concave Densitiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Global rates of convergence in log-concave density estimation

Kim¹,

Samworth²

2016

Ann. Statist.

109

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“…In particular we will review some of the key results for these classes in the next section. For bounds on densities of s —concave distributions on ℝ see Doss and Wellner [2013]; for probability tail bounds for s —concave measures on ℝ d , see Bobkov and Ledoux [2009]. For moment bounds and concentration inequalities for s —concave distributions with s < 0 see Adamczak et al [2012] and Guédon [2012], section 3.…”

Section: Log-concavity and Strong Log-concavity: Definitions And Bmentioning

confidence: 99%

“…They show that the MLE exists and is Hellinger consistent. Doss and Wellner [2013] have obtained Hellinger rates of convergence for the maximum likelihood estimators of log-concave and s –concave densities on ℝ, while Kim and Samworth [2014] study Hellinger rates of convergence for the MLEs of log-concave densities on ℝ d . Henningsson and Astrom [2006] consider replacement of Gaussian errors by log-concave error distributions in the context of the Kalman filter.…”

Section: Some Open Problems and Further Connections With Log-concamentioning

confidence: 99%

Log-concavity and strong log-concavity: A review

Saumard¹,

Wellner²

2014

Statist. Surv.

Self Cite

205

154

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We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

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“…Their nonparametric maximum likelihood estimators were studied by Dümbgen & Rufibach (2009), Cule et al (2010), Cule & Samworth (2010), Chen & Samworth (2013), Pal et al (2007) and Dümbgen et al (2011) (referred as [DSS 2011] thereafter). The convergence rates of these estimators for log-concave densities were studied by Doss & Wellner (2013) and Kim & Samworth (2014). Such estimators provide more generality and flexibility without any tuning parameters.…”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation of the mixture of log-concave densities

Yao

2016

Computational Statistics & Data Analysis

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Finite mixture models are useful tools and can be estimated via the EM algorithm. A main drawback is the strong parametric assumption about the component densities. In this paper, a much more flexible mixture model is considered, which assumes each component density to be log-concave. Under fairly general conditions, the log-concave maximum likelihood estimator (LCMLE) exists and is consistent. Numeric examples are also made to demonstrate that the LCMLE improves the clustering results while comparing with the traditional MLE for parametric mixture models.

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Global rates of convergence of the MLEs of log-concave and $s$-concave densities

Cited by 56 publications

References 35 publications

Global rates of convergence in log-concave density estimation

Global rates of convergence in log-concave density estimation

Log-concavity and strong log-concavity: A review

Maximum likelihood estimation of the mixture of log-concave densities

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