Computing maximum likelihood estimators of a log-concave density function

Abstract

Abstract 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. Keywords: Non-parametric density estimationShape restrictionInterior point MethodIterative convex minorant algorithmNewton method on subspace Additional informationNotes on contributorsKaspar Rufibach Email: [email protected]