Computing maximum likelihood estimators of a log-concave density function

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 estimation
• Shape restriction
• Interior point Method
• Iterative convex minorant algorithm
• Newton method on subspace

Additional information

Notes on contributors

Kaspar Rufibach

Email: [email protected]