Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

Duembgen, Lutz; Rufibach, Kaspar

doi:10.3150/08-bej141

Cited by 139 publications

(239 citation statements)

References 34 publications

Supporting

Mentioning

231

Contrasting

Order By: Relevance

“…Shape-constrained maximum likelihood dates back to Grenander (1956), who treated monotone densities in the context of mortality data. Recently there has been considerable interest in alternative shape constraints, including convexity, k-monotonicity and log-concavity (Groeneboom et al 2001;Dümbgen and Rufibach 2008;Balabdaoui and Wellner 2007). However, these works have all focused on the case of univariate data.…”

Section: Log-concave Density Estimationmentioning

confidence: 99%

“…1-dimensional log-concave density estimation via maximum likelihood is discussed in Dümbgen and Rufibach (2008); computational aspects are treated in Rufibach (2007). It is in the multivariate case, however, where kernel density estimation is more difficult and parametric models less obvious, where a log-concave model may be most useful.…”

Section: Log-concave Densitiesmentioning

confidence: 99%

See 1 more Smart Citation

LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

Cule¹,

Gramacy²,

Samworth³

2009

J. Stat. Soft.

View full text Add to dashboard Cite

In this article we introduce the R package LogConcDEAD (Log-concave density estimation in arbitrary dimensions). Its main function is to compute the nonparametric maximum likelihood estimator of a log-concave density. Functions for plotting, sampling from the density estimate and evaluating the density estimate are provided. All of the functions available in the package are illustrated using simple, reproducible examples with simulated data.

show abstract

Section: Log-concave Density Estimationmentioning

confidence: 99%

Section: Log-concave Densitiesmentioning

confidence: 99%

LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

Cule¹,

Gramacy²,

Samworth³

2009

J. Stat. Soft.

View full text Add to dashboard Cite

show abstract

“…For general d, a slower, non-smooth optimisation method based on Shor's r-algorithm is implemented in the R package LogConcDEAD (Cule et al, 2007;Cule, Gramacy and Samworth, 2009); see also Koenker and Mizera (2010) for an alternative approximation approach based on interior point methods. On the theoretical side, through a series of papers (Pal, Woodroofe, and Meyer, 2007;Dümbgen and Rufibach, 2009;Seregin and Wellner, 2010;Schuhmacher and Dümbgen, 2010;, we now have a fairly complete understanding of the global consistency properties of the log-concave maximum likelihood estimator (even under model misspecification).…”

Section: Introductionmentioning

confidence: 99%

Global rates of convergence in log-concave density estimation

Kim¹,

Samworth²

2016

Ann. Statist.

109

View full text Add to dashboard Cite

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.

show abstract

“…Specifically, it seems to be relatively little studied if it is feasible that the sample be generated by a log-concave distribution. See, for example [8,9].…”

Section: Introductionmentioning

confidence: 99%

Distinguishing Log-Concavity from Heavy Tails

Asmussen

Lehtomaa

2017

Risks

View full text Add to dashboard Cite

Well-behaved densities are typically log-convex with heavy tails and log-concave with light ones. We discuss a benchmark for distinguishing between the two cases, based on the observation that large values of a sum X 1 + X 2 occur as result of a single big jump with heavy tails whereas X 1 , X 2 are of equal order of magnitude in the light-tailed case. The method is based on the ratio |X 1 − X 2 |/(X 1 + X 2 ), for which sharp asymptotic results are presented as well as a visual tool for distinguishing between the two cases. The study supplements modern non-parametric density estimation methods where log-concavity plays a main role, as well as heavy-tailed diagnostics such as the mean excess plot.

show abstract

Maximum likelihood estimation of a log-concave density and its distribution function: Basic properties and uniform consistency

Cited by 139 publications

References 34 publications

LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

Global rates of convergence in log-concave density estimation

Distinguishing Log-Concavity from Heavy Tails

Contact Info

Product

Resources

About