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Quasi-concave density estimation. (English) Zbl 1200.62031
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 A. Rényi entropy estimators [Proc. 4 th Berkeley Symp. Math. Stat. Probab. 1, 547–561 (1961; Zbl 0106.33001)] 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.

MSC:
62G07 Density estimation
62H12 Estimation in multivariate analysis
90C25 Convex programming
62B10 Statistical aspects of information-theoretic topics
62G20 Asymptotic properties of nonparametric inference
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