# Log-concave density estimation

### Abstract

We study nonparametric maximum likelihood estimation of a univariate log-concave probability density. Many parametric models feature log-concavity (at least for certain parameter ranges), e.g. normal, gamma, extreme value, laplace or logistic. Note further that log-concavity entails unimodality. The maximum likelihood estimator can be constructed to be the solution of a linearly constrained optimization problem. We provide algorithms using interior point and other methods to solve this optimization task. Some general properties and two entirely different characterizations of the density estimator are derived. It is shown that its rate of convergence with respect to supremum norm on a compact interval is at least $(log(n)/n)^{1/3}$ and typically $(log(n)/n)^{2/5}$. This entails that the corresponding distribution function estimator is essentially equivalent to the empirical cumulative distribution function. Furthermore, a general property of log-concave densities enables one to define a simple plug-in monotone hazard rate estimator.

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Palo Alto, United States