QUASI-CONCAVE DENSITY ESTIMATION

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 restricti...

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Bibliographic Details
Published in:The Annals of statistics 2010-10, Vol.38 (5), p.2998-3027
Main Authors: Koenker, Roger, Mizera, Ivan
Format: Article
Language:English
Subjects:
62B10
62G05
62G07
62H12
90C25
94A17
Concavity
Convex analysis
convex optimization
Convexity
Density
Density estimation
duality
Entropy
Estimators
Exact sciences and technology
General topics
Mathematical functions
Mathematics
Maximum likelihood estimation
Maximum likelihood estimators
Nonparametric inference
Objective functions
Probability
Probability and statistics
Sciences and techniques of general use
semidefinite programming
shape constraints
Statistics
strongly unimodal
Studies
unimodal
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/10-AOS814