QUASI-CONCAVE DENSITY ESTIMATION

By Roger W. Koenker, Ivan Mizera. 2010

87 citations recorded by the source corpus.

Maximum likelihood estimation of a log-concave probability density is formulated as a convex optimization problem and shown to have an equivalent dual formulation as a constrained maximum Shannon entropy problem. Closely related maximum Renyi entropy estimators that impose weaker concavity restrictions on the fitted density are also considered, notably a minimum Hellinger discrepancy estimator that constrains the reciprocal of the square-root of the density to be concave. A limiting form of these estimators constrains solutions to the class of quasi-concave densities.

Subfields: Electrical and Electronic Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/57d48db7f81569e3281adaa3dd4d1856ecfd5f10

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Citations in the corpus

Inference and Modeling with Log-concave Distributions (2009, methodology): “An alternative way to compute the MLE with convex programming algorithms is described in Koenker and Mizera (2008).”
Maximum likelihood estimation of a multi‐dimensional log‐concave density (2008, background): “Other recent related work includes Seregin and Wellner (2010), Schuhmacher et al. (2009), Schuhmacher and Dümbgen (2010) and Koenker and Mizera (2010).”
Recent Progress in Log-Concave Density Estimation (2017, methodology): “Results on the maximum likelihood estimator in the case d = 1 are recently available (Doss and Wellner, 2016a), but estimation techniques based on Rényi divergences are also attractive here (Koenker and Mizera, 2010; Han and Wellner, 2016a).”
Fast nonnegative deconvolution for spike train inference from population calcium imaging. (2009, methodology): “Perhaps the best approximation would use the closest log-concave relaxation to the Poisson model [45].”
Global rates of convergence in log-concave density estimation (2014, methodology): “ 2011). 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.”
Variable selection with error control: another look at stability selection (2011, methodology): “insensitivity of the output of the algorithm to variations in the training set; such studies include Lange et al. (2003), Kalousis et al.”
APPROXIMATION BY LOG-CONCAVE DISTRIBUTIONS, WITH APPLICATIONS TO REGRESSION (2010, background): “Extensions of one or more of these aspects to the multivariate setting are presented by Cule, Samworth and Stewart (2010), Cule and Samworth (2010), Koenker and Mizera (2010), Seregin and Wellner (2010) and Schuhmacher and Dümbgen (2010).”