Computing maximum likelihood estimators of a log-concave density function

We consider the problem of estimating a density function that is assumed to be log- concave. This semi-parametric model includes many well- known parametric classes; such as Normal, Gamma, Laplace, Logistic, Beta or Extreme value distributions, for specific parameter ranges. It is known that the maximum likelihood estimator for the log- density is always a piecewise linear function with at most as many knots as observations, but typically much less. We show that this property can be exploited to design a linearly constrained optimization problem whose iteratively calculated solution yields the estimator. We compare several standard and one recently proposed algorithm regarding their performance on this problem.